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[Assistant] After all, in the very casualness of Gatsby's party.


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[Assistant] There were romantic possibilities totally absent from her world.


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[Assistant] What was it up there in that song that seemed to be calling her back inside? What would happen now in the dim incalculable hours? Perhaps some unbelievable guest would arrive.


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[Assistant] A person infinitely rare and to be marveled at. Some authentically radiant young girl who with one fresh glance at Gatsby, one moment of magical encounter, would blot out those five years of unwavering devotion. I stayed late that night. Gatsby asked me to wait until he was free.


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[Assistant] And I lingered in the garden until the inevitable swimming party had run up, chilled and exalted from the black beaches.


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[Assistant] Until the lights were extinguished in the guest rooms overhead.


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[Assistant] When he came down the steps at last, the tanned skin was drawn unusually tight on his face, and his eyes were bright and tired. She didn't like it, he said immediately. Of course she did. She didn't like it, he insisted. She didn't have a good time. He was silent, and I guessed at his unutterable depression. I feel far away from her, he said. It's hard to make her understand. You mean about the dance?


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[Assistant] He dismissed all the dances he had given with a snap of his fingers.


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[Assistant] He wanted nothing less of Daisy than that she should go to Tom and say, I never loved you.


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[Assistant] After she had obliterated four years with that sentence, they could decide upon the more practical measures to be taken. One of them was that after she was free, they were to go back to Louisville and be married from her house, just as if it were five years ago.


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[Assistant] He broke off and began to walk up and down a desolate path of fruit rinds and discarded favors and crushed flowers.


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[Assistant] I wouldn't ask too much of her, I ventured. You can't repeat the past. Can't repeat the past? He cried incredulously. Why, of course you can.


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[Assistant] He looked around him, wildly, as if the past were lurking here and the shadow of his house just out of reach of his hand.


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[Assistant] I'm going to fix everything just the way it was before, he said, nodding determinedly. She'll see. He talked a lot about the past.


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[Assistant] And I gather that he wanted to recover something, some idea of himself perhaps, that had gone into loving Daisy. His life had been confused and disordered since then.


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[Assistant] But if he could once return to a certain starting place and go over it all slowly, he could find out what that thing was.


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[Assistant] One autumn night five years before they had been walking down the street when the leaves were falling.


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[Assistant] And they came to a place where there were no trees and the sidewalk was white with moonlight. They stopped here and turned toward each other.


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[Assistant] Now it was a cool night with that mysterious excitement in it which comes at the two changes of the year.


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[Assistant] Out of the corner of his eye Gatsby saw that the blocks of the sidewalks really formed a ladder.


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[Assistant] And mounted to a secret place above the trees. He could climb to it if he climbed alone.


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[Assistant] And once there he could suck on the pap of life, gulp down the incomparable milk of wonder.


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[Assistant] His heart beat faster and faster as Daisy's white face came up to his own. He knew that when he kissed this girl and forever wed his unutterable visions to her perishable breath, his mind would never romp again like the mind of God.


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[Assistant] So he waited listening for a moment longer to the tuning fork that had been struck upon a star. Then he kissed her.


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[Assistant] Even through his appalling sentimentality, I was reminded of something. An illusive rhythm.


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[Assistant] A fragment of lost words that I had heard somewhere a long time ago.


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[Assistant] For a moment a phrase tried to take shape in my mouth and my lips parted like a dumb man.


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[Assistant] As though there was more struggling upon them than a wisp of startle there, but they made no sound.


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[Assistant] And what I had almost remembered was uncommunicable forever.


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[Assistant] It was when curiosity about Gatsby was at its highest.


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[Assistant] And as obscurely as it had begun, his career as Trimalchio was over.


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[Assistant] Only gradually did I become aware that the automobiles which turned expectantly into his drive stayed for just a minute and then drove sulkily away. Wondering if he was sick, I went over to find out.


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[Assistant] An unfamiliar butler with a villainous face squinted at me suspiciously from the door. Is Mr. Gatsby sick? Nope. After a pause he added, Sir, in a dilatory grudging way.


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[Assistant] I hadn't seen him around and I was rather worried. Tell him Mr. Carraway came over. Who? He demanded rudely. Carraway. Carraway. All right, I'll tell him. Abruptly he slammed the door.


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[Assistant] My fin informed me that Gatsby had dismissed every servant in his house a week ago.


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[Assistant] The grocery boy reported that the kitchen looked like a pigsty, and the general opinion in the village was that the new people weren't servants at all. Next day Gatsby called me on the phone. Going away? I inquired. No, old sport. I hear you fired all your servants. I wanted somebody who wouldn't gossip. Daisy comes over quite often, in the afternoon.


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[Assistant] So the whole caravanserai had fallen in like a card house at the disapproval in her eyes. There some people Wolfsheim wanted to do something for.


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[Assistant] They're all brothers and sisters. They used to run a small hotel. I see.


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[Assistant] He was calling up at Daisy's request. Would I come to lunch at her house tomorrow? Miss Baker would be there.


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[Assistant] Half an hour later, Daisy herself telephoned and seemed relieved to find that I was coming. Something was up.


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[Assistant] And yet I couldn't believe that they would choose this occasion for a scene.


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[Assistant] Especially for the rather harrowing scene that Gatsby had outlined in the garden. The next day was broiling, almost the last, certainly the warmest of the summer.


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[Assistant] As my train emerged from the tunnel into sunlight, only the hot whistles of the national biscuit company broke the simmering hush at noon.


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[Assistant] The straw seats of the car are hovered on the edge of combustion.


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[Assistant] The woman next to me perspired delicately for a while into her white shirt waist, and then, as her newspaper dampened under her fingers,


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[Assistant] Lapsed despairingly into deep heat with a desolate cry, her pocketbook slapped to the floor. Oh my! she gasped.


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[Assistant] I picked it up with a weary bend and handed it back to her, holding it at arm's length and by the extreme tip of the corners to indicate that I had no designs upon it, but everyone nearby, including the woman, suspected me just the same. Hot, said the conductor to familiar faces.


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[Assistant] My commutation ticket came back to me with a dark stain from his hand.


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[Assistant] That anyone should care in this heat whose flushed lips he kissed.


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[Assistant] Who's head may damp the pajama pocket over his heart? Through the hall of the Buchanan's house blew a faint wind.


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[Assistant] Carrying the sound of the telephone bell out to Gatsby and me as we waited at the door.


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[Assistant] I'm sorry madam, but we can't furnish it, it's far too hot to touch this noon. What he really said was, yes, yes, I'll see.


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[Assistant] Madame, expect you in the salon, he cried, needlessly indicating the direction. In this heat, every extra gesture was in the front of the common store of life.


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[Assistant] The room shadowed well with awnings was dark and cool.


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[Assistant] Daisy and Jordan lay upon an enormous couch like silver idols weighing down their own white dresses against the singing breeze of the fans. We can't move, they said together.


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[Assistant] Jordan's fingers powdered white over their tan, rested for a moment in mind.


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[Assistant] And Mr. Thomas Buchanan, the athlete I inquired.


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[Assistant] Simultaneously I heard his voice, gruff, muffled, husky, at the hall telephone.


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[Assistant] Gatsby stood in the center of the crimson carpet and gazed around with fascinated eyes. Daisy watched him and laughed her sweet, exciting laugh. A tiny gust of powder rose from her bosom into the air. The rumor is, whispered Jordan, that that's Tom's girl on the telephone. We were silent.


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[Assistant] And as for your bothering me about it at lunchtime, I won't stand that at all. Holding down the receiver, said Daisy cynically.


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[Assistant] Tom flung open the door, blocked out its space for a moment with his thick body, and hurried into the room.


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[Assistant] As he left the room again, she got up and went over to Gatsby and pulled his face down, kissing him on the mouth. You know I love you, she murmured. You forget there's a lady present, said Jordan. Daisy looked around doubtfully. You kiss Nick, too. What a low, vulgar girl.


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[Assistant] I don't care, cried Daisy, and began to clog on the brick fireplace.


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[Assistant] Then she remembered the heat and sat down guiltily on the couch, just as a freshly laundered nurse leading a little girl came into the room. Blessed, precious, she crooned, holding out her arms. Come to your own mother that loves you.


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[Assistant] The child, relinquished by the nurse, rushed across the room and rooted shyly into her mother's dress.


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[Assistant] The Blessed Precious. Did Mother get powder on your old yellowy hair? Stand up now and say howdy-doo.


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[Assistant] Gatsby and I in turn leaned down and took the small reluctant hand.


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[Assistant] Afterward he kept looking at the child with surprise. I don't think he had ever really believed in its existence before.


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[Assistant] I got dressed before luncheon, said the child, turning eagerly to Daisy. That's because your mother wanted to show you off.


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[Assistant] You dream, you, you absolute little dream. Yes, admitted the child calmly. Aunt Jordan's got on a white dress too.


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[Assistant] How do you like mother's friends? Daisy turned her around so that she faced Gatsby. Do you think they're pretty? Where's daddy?


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[Assistant] She doesn't look like her father, explained Daisy. She looks like me. She's got my hair in shape of her face. Daisy sat back upon the couch. The nurse took a step forward and held out her hand. Come, Pammy. Goodbye, sweetheart.


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[Assistant] With a reluctant backward glance, the well-disciplined child held to her nurse's hand


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[Assistant] They certainly look cool, he said with visible tension. We drank in long, greedy swallows.


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[Assistant] I read somewhere that the sun's getting hotter every year, said Tom genially. It seems that pretty soon the earth's going to fall into the sun.


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[Assistant] Oh, wait a minute, it's just the opposite. The sun's getting colder every year.


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[Assistant] Come outside, he suggested to Gatsby. I'd like you to have a look at the place. I went with them out to the veranda.


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[Assistant] On the green sands stagnant in the heat, one small sail crawled slowly toward the fresher sea.


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[Assistant] The anxiety from the unexpected shared elevator ride addled my brain. So when Miranda asked whose shoes I was wearing I thought perhaps she thought I was not wearing my own. Uhm, mine?


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[Assistant] I said, without realizing until the words had been spoken that it sounded not only disrespectful, but downright obnoxious.


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[Assistant] The gaggle of clackers began to twitter until Miranda turned her wrath on them. You, she said crisply to the cultish new girl, who was probably seeing Miranda for the first time.


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[Assistant] The girl hung her head and opened her mouth to apologize but Miranda barreled on.


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[Assistant] And you, she said, walking over and standing directly in front of Jocelyn, the highest ranking among them and a favorite of all the editors.


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[Assistant] You think there aren't a million girls who want your job and who understand couture just as well as you?


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[Assistant] She took a step back, slowly moved her eyes up and down each of their bodies


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[Assistant] Lingering just long enough to make each feel fat, ugly, and inappropriately clad, and commanded them all to return to their desks.


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[Assistant] They nodded their heads furiously while they moved quickly back to the fashion area. It wasn't until they'd all left that I realized we were alone. Again. Andrea, I won't tolerate being spoken to that way by my assistant.


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[Assistant] She declared, walking toward the door that would lead us to the hallway.


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[Assistant] Miranda, I- Enough! She paused at the door and looked at me. Whose shoes are you wearing? She asked again. I checked out my black sling bags and wondered how to tell the most stylish woman in the western hemisphere that I was wearing a pair of shoes I'd purchased at Ann Taylor Loft. I bought them in Spain. I said quickly.


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[Assistant] It was at some adorable boutique in Barcelona right off Las Ramblas that carried this new Spanish designer's line.


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[Assistant] She folded her hand into a fist, put it over her mouth and cocked her head.


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[Assistant] Andrea, they're unacceptable. Find some decent footwear in the closet and get me a coffee. She looked at me and looked at the door, and I understood I was to reach forward and open it for her, which I did.


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[Assistant] Yes, Miranda? I stopped in my tracks and turned to face her. I expect the restaurant review I asked you for is on my desk? Uhm, well, actually, I've had a little trouble locating it. I've spoken to all the papers and it seems none of them have run a review of an Asian Fusion restaurant in the past few days. Do you, uhm, happen to remember the name of the restaurant?


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[Assistant] Without realizing it, I was holding my breath and bracing for the onslaught.


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[Assistant] I'd spoken to their restaurant reviewer just that morning, and he had sworn there were no reviews that fit my description. She was cracking up for sure, and I was the one who was going to get blamed. The coffee run only took a few minutes since it was midday.


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[Assistant] I spent the rest of the afternoon arguing with every assistant and editor I could reach at the New York Post.


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[Assistant] Ma'am, I've told you a dozen times, and I'll tell you again, we did not review any such restaurant.


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[Assistant] I know Ms. Priestley is a crazy woman, and I don't doubt that she's making your life a living hell.


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[Assistant] But I just can't produce an article that doesn't exist. Do you understand?


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[Assistant] This had come from an associate editor who had been assigned the task of finding my article to shut me up. Emily! Miranda called from inside her office. Yes, Miranda. We both answered. Emily, I can hear that you just spoke to the people at the post. She said, directing her attention in my direction. The real Emily looked relieved and sat down. Yes, Miranda, I just hung up with them. I've actually spoken to three different people there.


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[Assistant] And all of them insist that they haven't reviewed a single new Asian fusion restaurant in Manhattan at any point in the last week. Maybe it was before then? Manhattan. She looked confused and pissed off all at once. Who said anything about Manhattan? It was my time to be confused. Andrea.


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[Assistant] I've told you at least five times now that the review was written about a new restaurant in Washington. Since I'll be there next week, I need you to make a reservation.


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[Assistant] She cocked her head and moved her lips into what can only be described as a wicked smile. Being the idiot she took me for, I again spoke without thinking.


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[Assistant] Oh Miranda, I'm fairly certain that the New York Post doesn't do reviews of restaurants in Washington. Is that supposed to be funny, Andrea? Is that your idea of having a sense of humor? Her smile had disappeared.


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[Assistant] And she was leaning forward in her seat looking like a hungry vulture that was impatiently circling its prey. Uhm, no Miranda, I just thought that- Andrea!


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[Assistant] As I've made clear a dozen times already, the review I'm looking for is in the Washington Post. You've heard of that little newspaper, right?


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[Assistant] Just like New York has the New York Times, Washington DC has its own paper too. You see how that works. I'll get it for you right away. I stayed it as calmly as I could and quietly walked out. Oh, and Andrea.


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[Assistant] My heart lurched and my stomach wondered if it could take another surprise.


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[Assistant] I expect you to attend the party tonight to greet the guests. That's all. Did I hear her correctly?


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[Assistant] I whispered to Emily, who could do nothing but nod in motion for me to come to her side of the suite. I was afraid of this. She whispered gravely, like a surgeon telling a patient's family member that they'd found something horrible upon opening the chest cavity.


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[Assistant] She can't be serious. It's four o'clock on Friday. The party starts at seven. It's black tie for Christ's sake. There's no way on earth she expects me to go.


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[Assistant] Oh, she's quite serious, Emily said, picking up the phone.


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[Assistant] I'll help you, okay? You find the review in the Washington Post and get her a copy.


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[Assistant] I'll get you a dress and everything else you need for tonight. Don't worry, we'll figure it out. She began rapid fire dialing and whispering urgent sounding instructions into the phone. I stood and stared but she waved her hand and I snapped back to reality.


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[Assistant] Go! she whispered, looking at me with a rare hint of sympathy. And I went.


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[Assistant] In the afternoon, our judge came over and introduced himself. He told us he'd be going with us that night.


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[Assistant] About sundown, we piled in our buggy and drove a few miles downriver.


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[Assistant] Old Dan walked off a few yards, stretched his body, and dragged his claws through the soft bottom soil.


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[Assistant] Opening his mouth, he let out one loud ball and then disappeared in the thick timber. Little Anne was right on his heels. We took off after them. Grandpa got nervous. He said to me, don't you think you ought to whoop to them? I told him to wait a little while. There would be plenty of time for whooping.


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[Assistant] He snorted and said he thought a hunter always whooped to his dogs. I do, Grandpa, I said, but not before they strike a trail. We walked on. Every now and then we would stop and listen.


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[Assistant] I could hear the loud snuffing of old Anne. Once we caught a glimpse of little Anne as she darted across an opening that was bathed in moonlight. She was as silent as a ghost and as quick as a flitting shadow. Papa said, it sure is a beautiful night for hunting.


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[Assistant] The judge said, you can't beat these Ozark Mountain Knights for beauty. I don't care where you go.


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[Assistant] Grandpa started to say something. His voice was drowned out by the bell-like cry of little Anne. In a whisper I said, Come on, Anne. Hurry and help her.


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[Assistant] I felt the blood tingling in my veins. That wonderful feeling that only a hunter knows crept over my body. Looking over at Grandpa, I said, now you can whoop.


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[Assistant] The coon was running upriver toward our campground. We turned and followed. I could tell by the dogs' voices that they were running side by side and were hot on the trail. Closing my eyes, I could almost see them running, bodies stretched to their fullest length, legs pounding up and down, white steam rolling from their hot breath in the frosty night.


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[Assistant] Grandpa got tangled up in some underbrush and lost his hat and spectacles.


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[Assistant] I told Papa we had better stay on our side of the river and keep going until we could hear them again. Twenty minutes later we heard them coming back. We stopped. I think they have crossed back to our side, I said.


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[Assistant] About that time we heard it again. The judge started laughing. I know now what it is, he said. Those hounds had run that coup right back by our camp. The noise we heard was the other hunters whooping to them. Everyone laughed.


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[Assistant] A few minutes later, I heard my dog's bawling treed. On reaching the tree, Papa ran his hand back under his coat. He pulled out Grandpa's gun.


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[Assistant] Now that's a funny looking gun, the judge said. It's a 410 gauge pistol, isn't it? It's a very thing for this kind of work, Papa said.


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[Assistant] You couldn't kill a coon with it if you tried, especially if you're using birdshot, all it will do is sting his hide a little. At the crack of the gun, the coon gave a loud squall and jumped. My dogs lost no time in killing him. We skinned the coon, and soon we're on our way again.


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[Assistant] The next time my dogs treed, they were across the river from us. Finding a ripple, we pulled off our shoes and started across.


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[Assistant] Grandpa very gingerly started picking his way.


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[Assistant] His tender old feet moved from one smooth rock to another.


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[Assistant] Everything was fine until we reached midstream where the current was much swifter. He stepped on a loose round rock. It rolled and down he went.


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[Assistant] As the cold river water touched his body, he let out a yell that could have been heard for miles. He looked so funny, we couldn't keep from laughing.


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[Assistant] Papa and the judge helped him to his feet, laughing every step of the way. We finally reached the other side.


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[Assistant] Grandpa kept going in his wet clothes until he reached the tree where the dogs were.


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[Assistant] After killing the coon, we built a large fire so grandpa could dry his clothes.


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[Assistant] He'd get up as close to the fire as he could and turn this way and that. He looked so funny standing there with his long underwear steaming. I started rolling with laughter. He looked over at me and snapped. What's so funny? I said, nothing. Well, why you laughing? He said.


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[Assistant] At this remark, Papa and the Judge laughed until their eyes watered. Mumbling and grumbling, Grandpa said,


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[Assistant] It's after three o'clock, he said. You think there are three in another one?


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[Assistant] As if to throw the words back in the judge's face, old Dan opened up. I stood up and whooped, Hoooowee! Get him, Dan, get him! Put him up a little tree!


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[Assistant] There was a mad scramble. Grandpa tried to put his britches on backwards. The judge and Papa ran over to help him with his shoes. Each one tried to put his shoe on the wrong foot. I was laughing so hard I could do nothing.


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[Assistant] A hundred yards from the fire I realized we had forgotten the coonskins. I ran back for them.


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[Assistant] My dogs had jumped the coon in swamp land. He tore out for the river bottoms. I could tell they were close to him by their fast balling. All at once their baying stopped. We stood still and listened.


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[Assistant] Old Dan balled tree a few more times and then stopped. Grandpa asked, what's happened? I told him the coon probably pulled some kind of trick.


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[Assistant] Coming up to my dogs, we saw they were working up and down an old rail fence. We stood and watched.


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[Assistant] Every now and then old Dan would rear up on a large hackberry tree that was standing about seven feet from the fence and ball treed. As yet little Anne had not balled the tree bark. We watched her.


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[Assistant] She was working everywhere. She climbed up on the rail fence and followed its zigzag course until she disappeared in the darkness. I told Papa I was sure the coon had walked the rail fence and in some way had fooled my dogs.


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[Assistant] Old Dan would keep coming back to the hackberry tree. He would rear up on it and ball-treed. We walked up to him, looking the tree over. We could see that the coon wasn't in it. The judge said, It looks like he has him fooled.


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[Assistant] Maybe you had better call them off, Grandpa said. We can go someplace else and hunt.


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[Assistant] We got to get one more coon, even if I have to treat myself. For some reason no one laughed at his remark. It's almost daylight, Papa said.


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[Assistant] Yes, that's what has me worried, I said. We don't have time to do anymore hunting. If we lose this one, we're beat.


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[Assistant] The blackberry tree has something to do with his trick but I don't know what.


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[Assistant] I wouldn't feel too badly if I were you. I've seen some of the very best hounds fooled by a smart old coon.


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[Assistant] Regardless of all the discouraging talk, the love and belief I had in my little red hounds never faltered.


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[Assistant] I could see them now and then leaping over old logs, tearing through the underbrush, sniffing and searching for the lost trail.


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[Assistant] My heart swelled with pride. I whooped, urging them on. In a low voice, the judge said, I'll say one thing, they don't give up easily.


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[Assistant] Birds began to chirp all around us. The sky took on a light gray color. Tiny dim stars were blinking the night away.


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[Assistant] It looks like we're beat, Papa said. It's getting daylight.


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[Assistant] At that moment, the loud, clear voice of a red bone hound balling tree rang through the river bottoms. It was the voice of little Anne sucking in a mouthful of air. I held it. I could feel my heart pounding against my ribs.


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[Assistant] I closed my eyes tight and gritted my teeth to keep the tears from coming. Let's go to them, Grandpa said. No, wait a minute, I said. Why, he asked. Wait till old Dan gets there, I said.


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[Assistant] It's daylight now, and if we walk up to the tree, the coon will jump out. It's hard to keep a coon in a tree after daylight. Let's wait until old Dan gets there. Then if he jumps, he won't have a chance to get away. The boy's right, the judge said. It's hard to keep a coon in a tree after daybreak. Just then we heard old Dan. His deep voice shattered the morning silence.


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[Assistant] Searching for the lost trail, he had crossed the fence and worked his way out into an old field. Turning around, we saw him coming. He was a red blur in the grey morning shadows. Coming to the rail fence and without breaking his stride, he raised his body into the air. About half way over and while still in the air, he bawled.


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[Assistant] Hitting the ground with a loud grunt, he ran past us. Everyone hooped to him. Ahead was a deep washout about ten feet wide. On the other side was a cane break. His long red body, stretched to its fullest length, seemed to float in the air as he sailed over it. We could hear the tall stalks rattling as he plowed his way through them.


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[Assistant] A bunch of sleepy snowbirds rose from the thick cane, flitted over, and settled in a row on the old rail fence. Nearing the tree we could see it was a tall sycamore, and there, high in the top, was the coon.


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[Assistant] Let's walk back to that old fence. I think I know how the old fella pulled his trick.


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[Assistant] Back at the fence, the judge stood and looked around for a few minutes, smiling. He said, Is that how he did it? How? Grandpa asked. Still smiling, the judge said, That old coon walked this rail fence.


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[Assistant] Coming even with the hackberry tree, he leaped up on its side and climbed up. Notice how thick the timber is around here.


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[Assistant] See that limb way up there in the top? The one that runs over and almost touches the sycamore? We saw it, he meant.


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[Assistant] The coon walked out on that limb, he said, leaped over and caught the sycamore limb.


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[Assistant] Repeating this over and over, from tree to tree, he worked his way far out into the river bottom.


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[Assistant] What I can't figure out is how that hound found him. Gazing at little Anne, he shook his head and said,


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[Assistant] I've been hunting coons and judging coon hunts for forty years, but I've never seen anything like that. He looked at me. Well son, he said, you have tied the leading teams. There's only one more night of eliminations.


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[Assistant] Even if some of them get more than three coons, you will still be in the run often. From what I've seen here tonight, you have a good chance of winning the cup.


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[Assistant] I knew that little Anne had scented the coon in the air the same as she had the ghost coon. I walked over and knelt down by her side. The things I wanted to say to her, I couldn't for the knot in my throat, but I'm sure she understood.


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[Assistant] Let's see. What was a beautiful sight, Grandpa asked?


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[Assistant] Last night those little red hounds brought that coo right through camp. The judge said,


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[Assistant] We figured they did when we heard the noise. Laughing the man said, We heard them when they ran up the other side of the river. Way up above here they crossed over. We could tell they were coming back so we doused all the fires and sure enough they came right through camp.


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[Assistant] Those two little hounds were fifty yards behind the coog running side by side. Boy, they were picking them up and laying them down and bawling every time their feet touched the ground. I tell you, it was the prettiest that I ever saw.


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[Assistant] When the judge started telling about the last coon little Anne had treed, I took my dogs over to our tent and fed and watered them.


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[Assistant] After they had had their fill, I gave them a good rub down with a piece of gunny sack. Taking them out to the buggy, I tied them up.


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[Assistant] I stood and watched while they twisted round in the hay, making their bed.


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[Assistant] That day I tried to get some sleep in our tent, but the soaking grandpa had taken in the river had given him a cold causing him to snore.


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[Assistant] I'd have sworn he rattled the paper sacks in our grocery boxes.


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[Assistant] Taking a blanket, I went out to my dogs. Little Anne had wriggled up as close to old Dan as she could. Prying them apart, I lay down between them and fell asleep.


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[Assistant] The last night of the eliminations turned out like the second night. None of the judges turned in more than two highs.


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[Assistant] That day, about noon, the owners of the other winning teams and I were called for a conference with the head judge. He said, gentlemen, the eliminations are over. Only three sets of hounds are left for the runoff. The winner of tonight's hunt will receive the gold cup.


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[Assistant] If there's a tie for the championship, naturally there will be another runoff. He shook hands with each of us and wished us good luck.


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[Assistant] Attention began to build up in the camp. Here and there, hunters were standing in small groups, talking. Others could be seen going in and out of tents, with rolls of money in their hands.


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[Assistant] Grandpa was the busiest one of all. His voice could be heard all over the camp. Men were looking at me and talking in low tones. I strutted like a turkey gobbler.


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[Assistant] That evening while we were having supper a hunter dropped by. He had a small box in his hand. Smiling he said,


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[Assistant] Everyone has agreed that we should have jackpot for the winner. I've been picked to do the collecting. Drampa said, you may as well leave it here now. Looking at me, the hunter said,


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[Assistant] Son, I think almost every man in this camp is hoping you win it. But it's not gonna be easy.


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[Assistant] You gonna be up against four of the finest hounds there are? Turning to my father, he said,


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[Assistant] One is almost as big as a barn. The other one isn't much bigger than a jackrabbit.


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[Assistant] Do you remember where we jumped the last coon in the swamp? Papa said yes.


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[Assistant] Well, the way I figure, more than one coon lives in that swamp, I said. It's a good place for them, as there are lots of crawfish and minnows in those potholes.


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[Assistant] If a hound jumps one there, he has a good chance to trim. Papa asked, why?


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[Assistant] It's a long way back to the river and about the same distance to the mountains, I said. Either way he runs, a dog can get pretty close to him, and so he would have to take to a tree.


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[Assistant] That evening we climbed into grandpa's buggy and headed for the swamp. It was dark by the time we reached it. Grandpa handed papa his gun saying


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[Assistant] I hope I get to shoot it a lot tonight, Papa said. Under my breath I said, I do too.


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[Assistant] After untying the ropes from my dogs, I held onto their collars for a minute. Pulling them up close, I knelt down and whispered, This is the last night. I know you'll do your best. They seemed to understand and tugged at their collars


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[Assistant] I can feel that they know this hunt is important. They know it just as well as you or I.


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[Assistant] It was little Anne who found the trail before the echo of her sharp cry had died away, old Dan's deep voice floated out of the swamp. Well, let's go, Papa said eagerly. No, let's wait a minute, I said.


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[Assistant] Grandpa asked. To see which way he's gonna run, I said.


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[Assistant] By the time we had circled the swamp, they were bawling treed. The judge said,


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[Assistant] I felt my father's hand on my shoulder looking at me. He smiled and nodded his head.


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[Assistant] Papa and I knew I had judged the coon perfectly. He didn't have time to reach the river or the mountains.


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[Assistant] My dogs had treed the coon in a tall ash which stood about fifty yards from the river.


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[Assistant] I knew the 50 yards had saved us a good hour because he could have pulled trick after trick if he had gotten in the water.


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[Assistant] We spied the coon in the topmost branches. At the crack of a gun, he ran far out on a limb and jumped.


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[Assistant] He landed in an old fallen tree top. He scooted through it. Coming out on the other side, he ran for the river. The tangled mass of limbs slowed my dogs, and they all but tore the tree top apart getting out of it. The coon was just one step ahead of them as they reached the river. We heard them hit the water.


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[Assistant] Running over, we stood and watched the fight. The coon was at home in the river. He crawled up on old Dan's head, trying to force him under. Before he could do it, little Ann reached up and pulled him off. In a scared voice, Papa said, that water looks deep to me.


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[Assistant] Why, you couldn't whip them off with a stick. There's no use for anyone to get scared. They know exactly what they're doing. I've seen this more times than one.


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[Assistant] Grandpa was scared and excited. He was jumping up and down, whooping and hollering. Papa raised the gun to aim. I jumped and grabbed his arm. Don't do that! I yelled. You're sure to hit one of my dogs!


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[Assistant] Round and round in the deep water the fight went on. The coon climbed on old Dan's head and sank his teeth in one of his long, tender ears. Old Dan bawled with pain. Little Ann swam in and caught one of the coon's hind legs in her mouth. She tried hard to pull him off. All three disappeared under the water. I held my breath. The water churned and boiled.


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[Assistant] All three came to the top about the same time. The coon was between the bank we were standing on and my dogs. He swam toward us.


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[Assistant] They caught him again just as he reached shore. He fought his way free and ran for a large sycamore. Old Dan caught him just as he started up. I knew that was the end of the fight.


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[Assistant] After it was all over and the coon had been skinned, grandpa said, I hope we don't have to go through that again tonight. For a while I sure thought your dogs were goners. The judge said, well, have you ever seen that? Look over there.


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[Assistant] Old Dan was standing perfectly still with eyes closed and head hanging down. Little Anne was licking at his cut and bleeding ears. She always does that, I said.


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[Assistant] If you'll watch when she gets done with him, he'll do the same for her.


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[Assistant] We stood and watched until they had finished doctoring each other. Then, trotting side by side, they disappeared in the darkness. We followed along, stopping now and then to listen.


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[Assistant] What does it mean to truly love another?


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[Assistant] There was a time in my life when I thought I knew the answer.


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[Assistant] It means that I'd care for Savannah more deeply than I cared for myself, and that we'd spend the rest of our lives together. It wouldn't have taken much.


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[Assistant] She once told me that the key to happiness was achievable dreams, and hers were nothing out of the ordinary. Marriage, family, the basics.


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[Assistant] It meant I'd have a steady job, the house with the white picket fence


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[Assistant] And a minivan or SUV big enough to haul our kids to school or to the dentist or off to soccer practice or piano recitals.


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[Assistant] I could imagine lying beside her in bed at the end of the day, holding her while we talked and laughed, lost in each other's arms. It doesn't sound so far-fetched, right? When two people love each other? That's what I thought, too.


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[Assistant] And while part of me still wants to believe it's possible, I know it's not going to happen. When I leave here again, I'll never come back.


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[Assistant] For now though, I'll sit on the hillside overlooking her ranch and wait for her to appear. She won't be able to see me, of course.


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[Assistant] In the army, you learned to blend into your surroundings, and I learned well.


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[Assistant] Because I had no desire to die in some backward foreign dump in the middle of the Iraqi desert.


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[Assistant] But I had to come back to this small North Carolina mountain town to find out what happened.


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[Assistant] When a person sets a thing in motion, there's a feeling of unease, almost an ache, until you learn the truth. But of this, I am certain. Savannah will never know I've been here today.


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[Assistant] Part of me aches at the thought of her being so close, yet so untouchable. But her story and mine are different now.


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[Assistant] It wasn't easy for me to accept this simple truth because there was a time when our stories were the same. But that was six years and two lifetimes ago.


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[Assistant] There are memories for both of us, of course, but I've learned that memories can have a physical, almost living presence.


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[Assistant] And in this, Savannah and I are different as well.


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[Assistant] If hers are stars in the nighttime sky.


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[Assistant] Mine are the haunted empty spaces in between.


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[Assistant] And unlike her, I've been burdened by questions I've asked myself a thousand times since the last time we were together. Why did I do it? And would I do it again? It was I, you see, who ended it.


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[Assistant] One night, as he was resting on a tree, he started to grumble as usual.


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[Assistant] Little crane, speak carefully. Do you really think being a butterfly would be better?


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[Assistant] I speak the truth and tell no lie. By morning you shall be a butterfly.


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[Assistant] The poor little butterfly fell into a deep sleep.


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[Assistant] He was very tired after the chase and scared that he had to face.


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[Assistant] It had been one of its tentacles, and the octopus wasn't happy about the fish playing with it.


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[Assistant] And from behind the bushes came the two hunters. They were very evil looking. And soon the lion started to feel afraid.


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[Assistant] The crane had learnt his lesson.


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[Assistant] Okay, so this is an important day, and Friday was an important day. I, uh, I hope you enjoyed Professor Schwarz.


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[Assistant] Terrific lecture as much as I did. You probably saw me taking notes like mad for the...


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[Assistant] Bring it to class, uh, Wednesday, hopefully.


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[Assistant] About, and that will give us a chance to review, uh,


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[Assistant] Stochastic gradient descent, the central algorithm of deep learning.


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[Assistant] Uh, and then this today is about the central, uh, structure.


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[Assistant] Of deep neural nets and some of you will know already what they, how they're connected, what the, what the, what the function, uh, f, the learning function, could you call it the learning function that, that's constructed.


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[Assistant] The whole system is aiming at constructing this, uh, function f, uh, which learns the training data and then applying it to the test data. And the miracle is that it does so well, uh, in, in practice. That's what has transformed, uh,


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[Assistant] Deep learning into such a important, uh, uh, application.


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[Assistant] This, the, Chapter 7 has been up for months on the, uh,


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[Assistant] Uh, math.mit.edu slash learning from data site, and I'll add it to Stellar because that's where you'll be looking for it. Okay, and then the second


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[Assistant] The back propagation, the way to compute the gradient, uh, I'll, uh, probably reach that idea today.


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[Assistant] You'll see what, it's the chain rule, but how is it organized? Okay, so, so what's the structure? What's the plan for deep neural nets? Good. Start, starting here. So what we have is training data.


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[Assistant] x1 to x, what should I use for the?


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[Assistant] Number of training, of samples that we have in the training, the training data, well, say D for data. Okay. And each vector,


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[Assistant] Those are called feature vectors. So, equals.


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[Assistant] So each one, each x is, uh, has like m features, so maybe, maybe my notation isn't, isn't so hot here.


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[Assistant] I have a whole lot of vectors. Let me not use the subscript for those right away. So vectors, feature vectors. And each vector has got maybe, shall we say, m features.


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[Assistant] Is to create, and, and we're going to classify. Maybe we're going to classify men and women. Or...


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[Assistant] Boys and girls, so our, let's make it a classification problem, just a binary. So the classification problem is, what should we say, minus 1 or 1, which is sort of,


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[Assistant] Or zero or one, or boy or girl, or cat or dog, or truck or car, or anyway, just two classes. Two, two class. So I'm just going to do two class classification.


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[Assistant] We know which class the training data is in. For each vector x, we know the right answer. So we want to create a function that gives the right answer. And then we'll use that function.


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[Assistant] On other data. People know, so let me write that down. Create a function f of x.


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[Assistant] When the, uh, classification is minus 1, and f of x should be positive when the classification is plus 1.


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[Assistant] And as we know, we don't necessarily have to get every x, every sample right. That may be overfitting. If there's some sample that's just truly weird.


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[Assistant] Uh, by getting that right, we're going to be looking for a truly weird, uh,


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[Assistant] Data in the test set, and that's not a good idea. We want the, we want the, the, uh,


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[Assistant] Rule that, we're trying to discover the rule that covers almost all cases, but not every crazy, weird case. Okay. So that's our job. To create a function f of x.


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[Assistant] That is correct on almost all of the training data.


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[Assistant] I don't know if you've looked at that, so I'm going to ask you. Playground at tensor flow.


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[Assistant] How many know that site or have met with it? Just, just a few. Okay.


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[Assistant] Okay, so it's not a very sophisticated site. It's got only four examples. Four examples and, uh, uh,


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[Assistant] And the, yeaah, so the, so one example is a whole lot of points.


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[Assistant] There are blue, B for blue, inside a bunch of points that are


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[Assistant] Another set that are O for orange. Orange, blue. Okay. So those are the two classes, orange and blue.


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[Assistant] So the points x and the feature vector is, is, the feature vector is here, is just the x, y, the coordinates.


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[Assistant] Features are the x, y coordinates of the points. And our job is to


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[Assistant] Find a function that's positive on these points and negative on those points.


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[Assistant] So there's a simple model problem. And I recommend, well, just partly, if you're an expert in deep learning, this is for children, but morally here,


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[Assistant] I certainly learned from playing in this playground. So you, you set the, you set the step size.


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[Assistant] Do you set it or does it set it? No, yeah, yeah, I guess you can change it. I haven't, I don't think I've changed it.


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[Assistant] Uh, what else do you set? Oh, you set the, uh,


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[Assistant] Uh, the nonlinear activation, the nonlinear.


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[Assistant] Activation function. Active, I'll say, function.


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[Assistant] And let me just go over here and say what function


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[Assistant] People now mostly use the activation function.


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[Assistant] Pronounced different ways. I don't know how we got into that crazy thing. For this function, that is 0 and x.


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[Assistant] So the function, ReLU as a function of x is the maximum, the larger of 0 and x. It produces, the point is it's not linear.


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[Assistant] And the point is that if we didn't allow non-linearity in here somewhere, we couldn't even solve this playground problem.


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[Assistant] Because if our classifiers were all linear classifiers, like support vector machines, I couldn't separate the blue from the orange with a plane.


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[Assistant] It's got to somehow create some nonlinear function, which is maybe the function is trying to be a good, a good function would be a function of r and theta maybe.


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[Assistant] Uh, maybe, uh, r minus phi.


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[Assistant] So maybe the distance out to that, let's suppose that distance is five.


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[Assistant] Then r minus 5 will be negative on the blues, because r is small, and r minus 5 will be positive on the oranges, because r is bigger, and therefore we'll have the right signs.


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[Assistant] Less than zero or greater than zero and, uh, it'll classify this, uh, this data, this training data. Yeaah. So, so it has to do that. This is not a hard one to do. There are, are,


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[Assistant] Uh, four examples, as I say, two are trivial. It, it solves, it finds a good function. Well, yeaah, I've forgotten they're so trivial. They've, uh, they don't, shouldn't be mentioned. And then this is the medium, uh, test. And then the hard test is when you have oranges, you have a sort of spiral of oranges,


[User] Please transcribe this text

[Assistant] And inside you have a spiral of blues. That was, uh, cooked up by a...


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[Assistant] The system is trying to find a function that's positive on one spiral and negative on the other spiral, and that takes quite a bit of time. Many, many epochs. I learned what an epoch is.


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[Assistant] Did you know what an epoch is? I, I didn't know whether it was just a fancy word for counting the steps in gradient descent. But it counts the, it counts the steps alright. But one epoch is the number of steps that matches the size of the training data. So if you have a million samples,


[User] Please transcribe this text

[Assistant] Where gradient, ordinary gradient descent, you would be doing a million, you'd have a million by million problem per step. Of course, stochastic gradient descent.


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[Assistant] Just does a mini-batch of 1 or 32 or something. But anyway, if we had, uhm,


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[Assistant] Yeah, so, so it's the number of, you have to do enough mini-batches so that the total number you've covered is the equivalent of one full run through the training data.


[User] Please transcribe this text

[Assistant] And that was an interesting point. Did you pick up that point that, uh, in stochastic gradient descent?


[User] Please transcribe this text

[Assistant] You could either do it, you could either do a mini-batch and then put them back in the soup. So with replacement. Or you could just put your data in some order from 1 to a zillion.


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[Assistant] So here's a first x and then more and more x's. And then, uh, just randomize the order. So you'd have to randomize the order for stochastic gradient descent to be


[User] Please transcribe this text

[Assistant] Reasonable. And then take a mini-batch and a mini-batch and a mini-batch and a mini-batch and when you get to the bottom you've finished one epoch.


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[Assistant] And then you'd probably randomize again maybe if you were, if you were doing it, if you wanted to live right, and go through the,


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[Assistant] The mini-batches again, and probably do a thousand times. Yeah.


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[Assistant] Anyway, so I haven't said yet what you do, what this f of x is like, but you can sort of see it on the screen because as it creates this function f, it kind of plots it. And what you see on the screen is the


[User] Please transcribe this text

[Assistant] Is the zero set for that function. So perfect would be for it to go through zero, if I had another color. Oh, I do have another color. Look, this is the first time the whole semester.


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[Assistant] Uh, blue has appeared, okay. So, so, if it, if the function was positive there, in the, in this part, on the blues, and negative outside that region, for the oranges, that would be just what we want, right?


[User] Please transcribe this text

[Assistant] That would be what this little playground site is creating. And on the screen you'll see it, you'll see this, this curve where it crosses zero.


[User] Please transcribe this text

[Assistant] So that curve where it crosses zero is supposed to separate the two sets.


[User] Please transcribe this text

[Assistant] One set is positive, one set is negative, where zero is in between. And the point is, it's not a straight line, because we've got this nonlinear function. This, this, uh, this is nonlinear. And it allows us to, uh,


[User] Please transcribe this text

[Assistant] To, to have functions like r, r minus 5, and, uh, so at, at 5,


[User] Please transcribe this text

[Assistant] That's where the function would be zero, and that, you'll see that on the screen. You might just...


[User] Please transcribe this text

[Assistant] Go to tensor, playground at tensorflow. Of course, tensorflow is the big, big system. This is the, the child's department. But it's, uh, I


[User] Please transcribe this text

[Assistant] Thought it was pretty good, and then on this site, you decide how many layers there will be.


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[Assistant] How many neurons in each layer? So you create the structure that I'm about to draw. And, uh, and you won't be able to


[User] Please transcribe this text

[Assistant] You get to solve this problem, to find a function f that learns that data, uh, without a number of layers and a number of neurons. If you don't give it enough, you'll see it struggling, but it won't.


[User] Please transcribe this text

[Assistant] It's creating, the zero set tries to follow this, but it gives up at some point.


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[Assistant] This one doesn't take too many layers, and the two trivial examples are just a few neurons do the job. Okay, so now, so that's a little comment on


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[Assistant] On one website. If others, if other, if you know other websites that I should know.


[User] Please transcribe this text

[Assistant] And should call attention to, could you send me an email? You know, I'm just not aware of everything that's out there.


[User] Please transcribe this text

[Assistant] Or if you know a good, uh, a good convolutional neural net, CNN, that is available to practice on, where you could give it the training set. That's what I'm talking about here.


[User] Please transcribe this text

[Assistant] I'd be glad to know, because I just don't know all that I should. Okay, so what does the function look like? Well, as I say,


[User] Please transcribe this text

[Assistant] Linear isn't going to do it, but linear is a very important part of it, of this function f. So the function f really has the form, uh, well,


[User] Please transcribe this text

[Assistant] Okay, so let me erase that now. Okay.


[User] Please transcribe this text

[Assistant] Let me make this simple. I'll just have that one layer and then I'll have the output. This will be the output layer.


[User] Please transcribe this text

[Assistant] And it's just going to be one number. So each of these, so I'm going to have a matrix A, one.


[User] Please transcribe this text

[Assistant] That takes me from this, A1 will be, this will be 5 by, no, 6 by 5. Because I want 6 outputs.


[User] Please transcribe this text

[Assistant] And 5 inputs, 6 by 5 matrix, so I have 30 weights to choose there. And, uh,


[User] Please transcribe this text

[Assistant] So the y that comes out is going to be y1 will be A1 times x0. So x0 is the feature vector with five components.


[User] Please transcribe this text

[Assistant] So that's a purely linear thing, but we also want a, uh,


[User] Please transcribe this text

[Assistant] An offset function, offset vector. So that's a vector. This is then, this is the y that's coming out has six components. The A1 is 6 by 5.


[User] Please transcribe this text

[Assistant] The x0 was 5 by 1, and then of course this is 6 by 1. So these are the, these are the weights. Well, these are, yeah, I'll call them all weights. Weights to compute.


[User] Please transcribe this text

[Assistant] So in this, so these are connected. The usual picture is to show these, all these connections. I'll just put in some of them.


[User] Please transcribe this text

[Assistant] 36 parameter. And then I'm going to close this. It's going to be a very shallow.


[User] Please transcribe this text

[Assistant] Uh, thing, so that will be just, uh,


[User] Please transcribe this text

[Assistant] Right, so we're just getting one output. So that's just a vector at this final.


[User] Please transcribe this text

[Assistant] Point, but the, of course, the whole idea of deep neural nets is that you have many layers.


[User] Please transcribe this text

[Assistant] So you have that 36 more realistically is in the tens of thousands, and you have it


[User] Please transcribe this text

[Assistant] Multiple times. And the idea seems to be that, uh,


[User] Please transcribe this text

[Assistant] The first layer that you can sort of separate what layer 1 learns.


[User] Please transcribe this text

[Assistant] About the data, and from what layer two learns about the data. Layer one, there's A1 apparently.


[User] Please transcribe this text

[Assistant] By just looking after the computation, this learned some, uh,


[User] Please transcribe this text

[Assistant] Some basic facts about the data. This learns the next, next A2, which would go in here, would learn more


[User] Please transcribe this text

[Assistant] Uh, detail, and then A3 would learn more detail, so we would have, uh, uh, a number of layers. And it's that construction that, uh, has made neural nets successful. But I haven't finished because right now it's only linear. Right now I just have, I'll call it A2 in here.


[User] Please transcribe this text

[Assistant] Right now I would just have a matrix multiplication apply A1 and then apply A2. But in between, there's a, there's a one by one action on each


[User] Please transcribe this text

[Assistant] So that function acts on that number to give that number back again or to give 0.


[User] Please transcribe this text

[Assistant] So in, in there, in there is ReLU, right, in this, in this.


[User] Please transcribe this text

[Assistant] It becomes ReLU on each six, six copies of ReLU acting on each of those six numbers. Right. So that really x1


[User] Please transcribe this text

[Assistant] Comes from y1 by applying ReLU to it.


[User] Please transcribe this text

[Assistant] Then that gives the x. So here are the y's from the linear part. And here are the x. That's y1. That's a vector y1. The linear from just the linear plus an affine map. Linear plus constant.


[User] Please transcribe this text

[Assistant] That's affine. Uh, and then the next step is component by component.


[User] Please transcribe this text

[Assistant] We apply this function and, uh,


[User] Please transcribe this text

[Assistant] We get x2, x1, yeaah. And then do it again and again and again. So do you see that the function, how do I describe now the function f of x?


[User] Please transcribe this text

[Assistant] At, which depends on the, on the weights, on the A's and B's.


[User] Please transcribe this text

[Assistant] Yeah, let me do this. So what, what, this is the function f of x. f of x is going to be f3, let's say, of f2 of f1 of x.


[User] Please transcribe this text

[Assistant] One, two, three parentheses, right? Okay, so it's a chain, you could say. F is a, what's the right word for a chain of functions? If I take a function of a function,


[User] Please transcribe this text

[Assistant] Which is the reason I use the word chain is that the chain rule gives the derivative. So a function of a function of a function, uhm, that's called composition. Composing function. So this is a composition.


[User] Please transcribe this text

[Assistant] I don't know if there's a standard symbol for starting with F1 and do some composition and do some composition. And now what are those separate F's? Those separate F's are the, are the, so the separate F's are the, uh,


[User] Please transcribe this text

[Assistant] F1 of a vector would be, includes the ReLU part, the nonlinear part of A1 x0 plus B1.


[User] Please transcribe this text

[Assistant] So two parts, you do the linear or affine map on your feature vector, and then you, component by component, you, uh, apply that nonlinear function.


[User] Please transcribe this text

[Assistant] And it took some years before that nonlinear function became a big favorite.


[User] Please transcribe this text

[Assistant] People imagined that it was better, it was important to have a smooth function. So the original functions were


[User] Please transcribe this text

[Assistant] But of course, it turned out that the experiment showed that this worked even better.


[User] Please transcribe this text

[Assistant] Yeah, so that would be F1, and then F2 would have the same form, and F3 would have the same form.


[User] Please transcribe this text

[Assistant] So maybe this had 36 weights, and the next one would have another number, and the next another number. So we're getting, you get quite complicated functions by composition, by


[User] Please transcribe this text

[Assistant] You know, like e to the sine of x, or e to the sine of the logarithm of x, or things like that. Pure math is s.


[User] Please transcribe this text

[Assistant] Could you get, what functions can you get? You know, try to think of them all. Now what kind of functions do we have here? What can I say about f of x as a function, as a math person? What, what kind of a function is it? So it's created out of


[User] Please transcribe this text

[Assistant] Matrices and vectors out of a linear or affine map, followed by a non-linear, by that particular non-linear function.


[User] Please transcribe this text

[Assistant] So what kind of a function is it? Well, I've written those words down up here.


[User] Please transcribe this text

[Assistant] That f of x is going to be a continuous piecewise linear function.


[User] Please transcribe this text

[Assistant] Because every step is continuous.


[User] Please transcribe this text

[Assistant] That's a continuous function. Linear functions are a continuous function, so we certainly have, we're taking a composition of continuous functions so it's continuous. And it's piecewise linear because part of it is linear and part of it is piecewise linear. So, uh, so this is some continuous piecewise linear function


[User] Please transcribe this text

[Assistant] Of the kind of chain we have here of linear followed by ReLU.


[User] Please transcribe this text

[Assistant] So here's my question. This is the question I'm going to ask.


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[Assistant] And by the way, back propagation is certainly going to come Wednesday rather than today. That's a major topic in itself. Uh, so let me keep going with this function. Could you get any function whatsoever this way?


[User] Please transcribe this text

[Assistant] Well no, you only get continuous piecewise linear function. That's a, that's an interesting case.


[User] Please transcribe this text

[Assistant] Let me just ask you, well, one of the exercises says if I took two continuous piecewise linear functions, do you have a, the next 20 minutes are an attempt to


[User] Please transcribe this text

[Assistant] Give us a picture of the graph of a piecewise linear function. So what, what, say in, say a function of two variables. So I have m equal to 2.


[User] Please transcribe this text

[Assistant] And I draw its graph. Okay, help me to draw its graph. So this will be a graph of f of x1, x2.


[User] Please transcribe this text

[Assistant] And it's going to be continuous and piecewise linear. So what does its graph look like? That's the question. What's the graph of a piecewise linear function look like? Well, it's got flat pieces, right? In between the change from


[User] Please transcribe this text

[Assistant] I do say piecewise, that means it's got different pieces, but within a piece, it's linear.


[User] Please transcribe this text

[Assistant] And the pieces fit next to, fit with each other because it's continuous. So I visualize the, the, uh, well it's like origami. This is the theory of origami. Almost.


[User] Please transcribe this text

[Assistant] Uh, so right, origami, you take a flat thing and you fold it along straight fold.


[User] Please transcribe this text

[Assistant] So, so what's different from origami? Maybe not much. Uh, well maybe origami allows more


[User] Please transcribe this text

[Assistant] Then we allow here. Origami would allow you to fold it up and over.


[User] Please transcribe this text

[Assistant] So origami would give you a multi-valued.


[User] Please transcribe this text

[Assistant] Thing because, you know, it's got a top and a bottom and other folds. This is just going out to infinity in flat pieces.


[User] Please transcribe this text

[Assistant] And the question will be, how many pieces? So let me ask you that question. How many pieces, uh, do I have? Uh, you see what I mean by a piece?


[User] Please transcribe this text

[Assistant] So, so I'm thinking of a graph that has these flat pieces and they're connected along straight, straight edges.


[User] Please transcribe this text

[Assistant] And those straight edges come from


[User] Please transcribe this text

[Assistant] From the ReLU operation. That, that, well that's got two pieces.


[User] Please transcribe this text

[Assistant] Actually, we could do in one d- in one d, we could count the number of pieces pretty easily.


[User] Please transcribe this text

[Assistant] So what would, what would be a piecewise linear? Let me put it over here on the side and erase it soon. Okay.


[User] Please transcribe this text

[Assistant] Straight pieces. Straight pieces.


[User] Please transcribe this text

[Assistant] Like so, yeaah, you've got the idea. It's a broken line type. Sometimes people say broken line, but I'm never sure that's a good description of this.


[User] Please transcribe this text

[Assistant] Piecewise linear continuous. So it's continuous because the pieces meet and, uh, it's piecewise linear, obviously. Okay, so that's the kind of picture I have for a function of one variable. Now, my question is,


[User] Please transcribe this text

[Assistant] As an aid to try to visualize this function in 2D is to see if we can count the pieces.


[User] Please transcribe this text

[Assistant] See if we can count the pieces. Yes, and that, for that, uh, so that's in this, uh, notes. I found it in a paper by, uh,


[User] Please transcribe this text

[Assistant] Five authors for a, for a, for a meeting and, uh,


[User] Please transcribe this text

[Assistant] So actually, the whole world of, uh, neural nets, uh, it's, it's the...


[User] Please transcribe this text

[Assistant] Conferences every couple of years that everybody prepares for, submits more than one paper. So it's kind of a piecewise linear conference. And those are the...


[User] Please transcribe this text

[Assistant] The big conferences, okay. So this is the back propagation section and I want to look at the, at the, uh, okay. So.


[User] Please transcribe this text

[Assistant] So this paper by Kleinberg and four others. Kleinberg is a, he's a computer science guy at Cornell.


[User] Please transcribe this text

[Assistant] He was a PhD from here in math and he's a very cool.


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[Assistant] And significant person on, uh, not on, not so much on neural networks as just, uh, the, the whole, uh, this whole part of computer science. Right. So anyway,


[User] Please transcribe this text

[Assistant] They and other people too have asked the same problem.


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[Assistant] Suppose you have, suppose I'm in two variables.


[User] Please transcribe this text

[Assistant] And suppose, so what are you imagining now for the surface?


[User] Please transcribe this text

[Assistant] Oh, the, the graph of, of, of this, of f of x and y, it has these, it has these lines, fold lines. Right? I'm thinking it has fold lines.


[User] Please transcribe this text

[Assistant] So I could start with a complete plane, and I fold it along one line. So now it's like ReLU. There's one half plane there going to a different half plane there. Everybody with it? And now I take that function.


[User] Please transcribe this text

[Assistant] That surface, which just has two parts, and I put in another fold. Okay, how many parts have I got now?


[User] Please transcribe this text

[Assistant] I think four. Am I right? Four parts? Yes. Because...


[User] Please transcribe this text

[Assistant] This will be different from this, because it was folded along that line.


[User] Please transcribe this text

[Assistant] So these will be four different pieces. They have the same value at the center there, and they match along the lines.


[User] Please transcribe this text

[Assistant] So the number of flat pieces is four for this. So that's with two-fold.


[User] Please transcribe this text

[Assistant] And now I just want to ask you, with m folds, how many pieces are there? Can I get up to three folds?


[User] Please transcribe this text

[Assistant] So let me, let me just use a, get a notation, maybe R. R.


[User] Please transcribe this text

[Assistant] Is the number of, number of flat pieces.


[User] Please transcribe this text

[Assistant] I'll fold it plain because the dimension was 2. I'll fold it, fold it in time.


[User] Please transcribe this text

[Assistant] This would be a central step in understanding how close


[User] Please transcribe this text

[Assistant] The function, you know, what freedom you have in the function f. For example.


[User] Please transcribe this text

[Assistant] Can you approximate any continuous function?


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[Assistant] By one of these functions f, by taking enough folds.


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[Assistant] Seems like the answer should be yes and it is yes. So that's the, that's one big, for pure math, that's one question. Is the, is this class of functions universal?


[User] Please transcribe this text

[Assistant] So the universality theorem would be to say that any, any function, sine x, whatever, could be approximated


[User] Please transcribe this text

[Assistant] As close as you like by one of these guys with enough fold. And over here, we're kinda making it more numerical. We're gonna count the number of pieces.


[User] Please transcribe this text

[Assistant] So just to see how quickly do they grow. So what happens here?


[User] Please transcribe this text

[Assistant] So I have four folds now, this, this, right now I have...


[User] Please transcribe this text

[Assistant] N equal 2. M is 2 here in this, in this picture.


[User] Please transcribe this text

[Assistant] And I'm trying to draw the surface, and here I've put in two, did I take n? Yeah, two folds. And now I'm going to go up to three folds. Okay, so let me fold it along that line. How many pieces have I got now?


[User] Please transcribe this text

[Assistant] Let's see. Can I count those pieces? Is it seven? So what is a formula? What if I do another fold?


[User] Please transcribe this text

[Assistant] What happened, what happened, yeaah, let's pretend we do another full amount, yeaah.


[User] Please transcribe this text

[Assistant] It's a kinda nice question, cause it asks you to visualize this thing. Okay.


[User] Please transcribe this text

[Assistant] Uhm, so what happened, how many, how many of those lines will be, if I put in a fourth line, how many, yeaah, how many, uh,


[User] Please transcribe this text

[Assistant] New folds do I create? That's kind of the question. And I'm assuming that fourth line


[User] Please transcribe this text

[Assistant] Doesn't go through any of these points. It's sort of in general position. Yeah.


[User] Please transcribe this text

[Assistant] So I put in a fourth line, da da da da, there it is. Okay, so what happened here? How many new ones did it create? How many new ones did it create?


[User] Please transcribe this text

[Assistant] Let me make that one green, because I'm distinguishing. That's the guy that's added.


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[Assistant] After the original. We had, we had seven. We had seven pieces.


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[Assistant] Was it seven? It was, wasn't it? One, two, three, four, five, six, seven. But now how many pieces have I got? Or how many, how many pieces did this, uh, did this new line create?


[User] Please transcribe this text

[Assistant] That, that's, we want to build it up using recursion. How many pieces did this new, new, well, this new line folded, created one new piece there, right?


[User] Please transcribe this text

[Assistant] One new piece there, one new piece there, one new piece there. So there are four new pieces. Okay. Uh, yes. So there's some formula, right? That's going to tell us that. And now, what would the next one create? Well, I've got, now I have one, two, three, four lines.


[User] Please transcribe this text

[Assistant] So now I'm going to put through a fifth line, and that'll create a whole bunch of pieces. I'm losing the thread of this argument, but you're onto it, right? Uh, yeaah. So any suggestions? Yeaah.


[User] Please transcribe this text

[Assistant] That's right. So there's a recursion formula that I want to know. And I learned it from Kleinberg's paper.


[User] Please transcribe this text

[Assistant] We have an addition to do, so the recursion will tell me how much it goes up each with each new function.


[User] Please transcribe this text

[Assistant] And then we have to add. Okay, so the recursion formula, let me write that down.


[User] Please transcribe this text

[Assistant] So this is r of n and m that I'd like to find a formula for. It's the number of flat pieces


[User] Please transcribe this text

[Assistant] In m, with a m dimensional surface, well we're taking m to be 2, and n folds. And let's just, uh, so, uh,


[User] Please transcribe this text

[Assistant] And equal 1, 2, 3. Let's write down the numbers we know. With one fold, how many pieces? Two. Good. So far, so good.


[User] Please transcribe this text

[Assistant] With one fold, there were two pieces. So this is the count r.


[User] Please transcribe this text

[Assistant] And then with two folds, how many, ooh, we've gone past that point, so.


[User] Please transcribe this text

[Assistant] Can we get back to just those two? Was it four? Okay, thanks.


[User] Please transcribe this text

[Assistant] Now when I put in that third fold, how many did I have? Without the green line yet. Seven? Was it seven?


[User] Please transcribe this text

[Assistant] So the question is, how many new ones did I create, I guess? So that line got chopped into that piece, that piece, that piece, that piece, four pieces for the new line, four pieces for the new line, and then each of those pieces, like


[User] Please transcribe this text

[Assistant] Added a, added a flat bit because it, that piece from here to here separated these two which were previously just one piece, one flat piece. They, I folded on that line, I folded on this, I folded there. I think it went up by 4 to 11.


[User] Please transcribe this text

[Assistant] So now we just have to guess a formula that gives those, matches those numbers. And then of course we really should guess it for any m and any n.


[User] Please transcribe this text

[Assistant] And I'll write down the formula that they found.


[User] Please transcribe this text

[Assistant] It's, it involves binomial numbers. Every, everything in the world involves binomial numbers. Because they satisfy every identity you could think of.


[User] Please transcribe this text

[Assistant] R of, of, with n folds, and we're in m dimensions, so we've really, in our thinking, had m equal to 2, but we should, you know, grow up and


[User] Please transcribe this text

[Assistant] Get m to be five-dimensional, so we have a five-dimensional, uh, let's not think about that. Okay, so.


[User] Please transcribe this text

[Assistant] So, so for m equal to 2, which is my picture, it's n0 plus n1 plus n2. And what are these? What does that n2 mean, for example? What's that? That's a binomial number. I don't know if you're...


[User] Please transcribe this text

[Assistant] Keen on binomial numbers. Some people, their whole lives go into binomial numbers. So, so it's something like, is it, is it n factorial divided by


[User] Please transcribe this text

[Assistant] And minus 2 factorial and 2 factorial.


[User] Please transcribe this text

[Assistant] I think that's what that number means. That's the binomial number. So this would be, I go up to


[User] Please transcribe this text

[Assistant] So at this point I'm hoping to get the answer seven, I think. Right now I've put in...


[User] Please transcribe this text

[Assistant] I've gone up to m equal to, I'm in m equal to


[User] Please transcribe this text

[Assistant] I've gone up to two, so I've, yeaah. So I think...


[User] Please transcribe this text

[Assistant] Uh, yeaah, I've obviously allowed for three, three cuts, and the R, when we had just three, was seven. So is that, is this, uh,


[User] Please transcribe this text

[Assistant] So, so this is now I'm taking n to be 3, and I'm hoping for hope, uh, answer is sept.


[User] Please transcribe this text

[Assistant] So what is three, what is three, uh, shall I add these three things? So what is three,


[User] Please transcribe this text

[Assistant] I'm ashamed to admit. Three choose two. Thanks. I knew there was a good way. So what is three choose two?


[User] Please transcribe this text

[Assistant] And 2 is in there already, so that'd be 6 over


[User] Please transcribe this text

[Assistant] 1 times 2, this would be 3, would that be 3?


[User] Please transcribe this text

[Assistant] How do you know that? You're probably right. Three, I think, yeaah. Oh yeaah, it's probably a theorem that if these add to three,


[User] Please transcribe this text

[Assistant] Yeah, so, so this, I'm doing n equals 3 here. Okay, so we, yeah, I agree, that's 3. And what about n choose 0? What's, that you have to live with 0 factorial.


[User] Please transcribe this text

[Assistant] Uh, but then zero factorial is by no means zero.


[User] Please transcribe this text

[Assistant] I remember when I was an undergraduate having a bet on that.


[User] Please transcribe this text

[Assistant] I won, but he didn't pay off. So, uh, yeah. So, so 3, this is 3 factorial over 3 factorial times 0 factorial. So it's 6 over 6 times 1. So it's 1. Yeah. So 1 and 3 and 3 makes 7. So that proves the formula.


[User] Please transcribe this text

[Assistant] Well, it doesn't quite prove the formula, but the way to prove it is by an induction. And I'll just, if you like this stuff, the recursion.


[User] Please transcribe this text

[Assistant] That you use induction on. Which is just what we did now. What we did here. So here comes in a number four.


[User] Please transcribe this text

[Assistant] And it cuts through, yeah, and then we just counted the four pieces there. So yeah, so let me just tell you what the...


[User] Please transcribe this text

[Assistant] R of n and m, the number we're looking for, is the number that we had with one less cut, so that's the previous count of flat pieces, plus the number of


[User] Please transcribe this text

[Assistant] Plus the number that was here was 4, the number of pieces that cut that. And that's r of n minus 1, m minus 1. Yeah. And yeah, I won't go further.


[User] Please transcribe this text

[Assistant] Times up, but this, that rule for recursion is proved in the, in the section 7.1.


[User] Please transcribe this text

[Assistant] Taken from the paper by Kleinberg, uh, and others. Yeah, so okay. I think this is...


[User] Please transcribe this text

[Assistant] I don't know what you feel. This, for me, this like gave me a better feeling that I was understanding what kind of functions we had here. And, uh,


[User] Please transcribe this text

[Assistant] Uhm, so then the question is to, with this family of functions, we want to,


[User] Please transcribe this text

[Assistant] Choose two, we want to choose the A's and the weights, the A's and the B's to give, uh, to, to match the training data. And so that we have a problem in, uh, in minimizing the total loss and we have a,


[User] Please transcribe this text

[Assistant] We have a gradient descent problem. So we have to find the gradient. So that's Wednesday's job.


[User] Please transcribe this text

[Assistant] Good, thank you very much. That's, that's, uh, 7.1 is done.


[User] Please transcribe this text

[Assistant] Listening Library presents Magic Treehouse Merlin Missions Book Number 9 Dragon of the Red Dawn This is Mary Pope Osborne


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[Assistant] Teddy made a walking motion with his fingers and pointed toward the Frog Creek woods. Annie nodded eagerly.


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[Assistant] And don't forget to bring the wand of Dianthus.


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[Assistant] Annie was standing on the front porch. Let's go, she said.


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[Assistant] Jack and Annie ran across their yard and dashed up the sidewalk. I wonder why they came for us said Annie. I wonder where we're going said Jack. I wonder everything said Annie.


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[Assistant] Jack and Annie crossed the street and hurried into the Frog Creek woods.


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[Assistant] The early March trees looked weary from winter, gray and brown with no leaves on them yet. Look, said Annie out of breath, they're waiting for us. Jack looked up.


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[Assistant] Teddy and Kathleen were waving from the window of the magic tree house. Jack grabbed the rope ladder and started up. Annie followed.


[User] Please transcribe this text

[Assistant] When Jack and Annie climbed inside the treehouse, they threw their arms around Teddy and Kathleen. We're so glad to see you, cried Annie. And we are happy to see you also, said Kathleen.


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[Assistant] The sea girl's lovely water blue eyes sparkled.


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[Assistant] Indeed, said Teddy, it has been too long. What's our mission this time? Asked Jack. Where's Merlin sending us? Teddy glanced at Kathleen. I fear Merlin does not even know we are here, said Teddy. We have come not at his bidding, but for his sake. What does that mean? Asked Jack. Merlin is not well, said Kathleen.


[User] Please transcribe this text

[Assistant] He complains that he is getting old and feeble and that life is full of sorrows. He does not eat or sleep. Oh no, said Annie. All of Camelot wishes to help him, said Teddy, but no one knows quite how. What can we do to help, asked Jack. Teddy picked up a book from the corner of the tree house.


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[Assistant] Throughout the ages, people all over the world have sought the secrets of happiness, he said.


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[Assistant] Morgan wants you to search for four of these secrets to share with Merlin. She believes that the first one might be found here.


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[Assistant] Jack took the book from Teddy. He read the title aloud. A Journey to Old Japan.


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[Assistant] Oh wow, we've been to Japan before, said Annie. Before we met you, Jack said to Teddy and Kathleen, we had an adventure with ninjas. Yes, Morgan told us, said Teddy.


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[Assistant] Since Merlin has fallen ill, she has taken on much of his work. You are the one, do you not? asked Teddy. Yep, said Jack.


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[Assistant] He reached into his backpack and took out the wand of Dianthus.


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[Assistant] The spiraled wand was shaped like a unicorn's horn. With the help of the wand, you will make your own magic, said Teddy. That's what Merlin said when he gave it to us, said Annie. But he didn't say how, said Jack. It is very simple, said Teddy. The wand has three rules. First, it only works for the good of others. The wand can never be used for selfish reasons. Second,


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[Assistant] The wand works only after you have tried your very hardest without its help, said Kathleen. Do not attempt to use its magic too quickly.


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[Assistant] And third, the wand only works with a command of five words, said Teddy. So you must choose your words carefully. Can we review all that, please? Asked Jack. Don't worry, I've got it, said Annie.


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[Assistant] We have to go. We have to help Merlin as soon as we can.


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[Assistant] And this book from Camelot's library will bring you back home to Frog Creek, said Teddy, after you have completed your mission.


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[Assistant] He picked up another book lying in a corner. It was the book about Pennsylvania that Jack and Annie had used on their first magic treehouse adventures. Thanks, said Jack. Goodbye, said Annie. Take good care of Merlin. We will try, said Kathleen. She and Teddy raised their magic rings to their lips.


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[Assistant] They whispered words too softly for Jack and Annie to hear, then blew on the rings.


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[Assistant] As they blew, the young sorcerers began to fade into the cool morning air. In a moment, they had disappeared completely. Silence filled the treehouse. Annie turned to Jack. Ready? She said. Jack nodded. He pointed to the cover of the Japan book. I wish we could go there, he said. The treehouse started to spin. It spun faster and faster than everything was still.


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[Assistant] Chapter 2. The Imperial Garden. Jack opened his eyes.


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[Assistant] Soft morning light shown across the floor of the treehouse.


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[Assistant] Jack and Annie were wearing brown baggy pants and brown silk robes with blue sashes.


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[Assistant] On their feet were stiff white socks and straw sandals. Jack's backpack had turned into a burlap bag. Are we wearing bath robes? asked Jack. I think they're called kimonos, said Annie. Oh, right, said Jack. Where'd we land exactly? Jack and Annie looked out the window.


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[Assistant] Below the tree house was a beautiful garden filled with cherry trees and long-leaved willows.


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[Assistant] In the 1600s, the Imperial Garden surrounded the Imperial Palace in the capital city of Japan. The city was called Edo.


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[Assistant] In the mid-1800s, its name was changed to Tokyo. Tokyo, said Annie. I've always wanted to go to Tokyo. Me too, said Jack. He read on.


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[Assistant] The late 1600s in Japan were years of peace and prosperity. Art and culture thrived.


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[Assistant] It also lists the different countries you traveled to. He read more.


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[Assistant] Anyone who did not have a passport was considered a spy and punished severely. Uh oh, said Annie, we don't have passports. Yeah, that's a problem, said Jack.


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[Assistant] Hey, what if we use the wand of Dianthus to make passports? said Annie. Good idea, said Jack. He peeked inside his bag. Good, the wand of Dianthus was there. Wait, wait, said Annie. We can't.


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[Assistant] Remember the rules. We can only use the magic wand for the good of others. Oh, right, said Jack. And we have to try our hardest before we use the wand, said Annie. We haven't tried anything yet, said Jack. I guess we should just start looking for a secret of happiness and hope no one catches us, said Annie. Shh, said Jack. Listen. A bell was ringing in the distance. The ringing grew louder.


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[Assistant] Then came the sound of horses. Jack and Annie crouched down. They raised their heads just high enough to peek out the window. Through the flowery tree branches, they saw a small procession coming through the garden. The man leading the procession was ringing a bell. Two men walked behind him, holding up banners. Behind them, four men rode slowly on horseback. They all wore baggy trousers and puffy shirts. Their heads were shaved, except for knots of black hair.


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[Assistant] Each had two swords, a long one and a short one, hanging from his belt.


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[Assistant] At the very end of the procession wrote a man in a billowing purple robe and a small purple hat.


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[Assistant] Red tassels hung from the bridle of his large black horse. Jack looked at their research book again. He found a picture that looked like the man on the black horse. He read the caption to himself.


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[Assistant] In the 1600s, the military ruler known as the Shogun lived in the center of the Imperial Garden in a palace that had hundreds of rooms. That last guy is a Shogun, Jack whispered to Annie. He lives in a big palace in the garden. He kept reading. Often the Shogun's warriors traveled with him. They were called samurai. Man, whispered Jack, those other guys are samurai.


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[Assistant] He and Annie had barely escaped an armored samurai on their earlier trip to Japan.


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[Assistant] Samurai were excellent horsemen, well trained in the arts of fighting. The code of the samurai was strict. Samurai did not show their feelings. They had great powers of concentration. They're gone, said Annie. Jack looked out the window.


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[Assistant] The shogun and his samurai warriors had disappeared down a tree-shaded dirt road. We should get out of the Imperial Garden fast, said Jack. If we stay here, we're just asking to be caught. How do we get out? asked Annie. Jack looked in the Japan book. He found a map of Edo. Look, he said, pointing at the map.


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[Assistant] We have to get over this bridge that leads away from the Imperial Garden into the city. The bridge is on the east side of the garden.


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[Assistant] The morning sun is over there, said Annie, squinting into the sunlight. So that must be east. Let's climb down and head that way. Good plan. Then we'll be walking in the opposite direction of those samurai, said Jack. Right, said Annie. She started down the rope ladder. Be careful, said Jack.


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[Assistant] Jack put the Japan book into his burlap bag and slung the bag over his shoulder.