#############################


# 4. Multivariate Regression


#############################

#################

## Answer 4(a)

#################

input-output training data $ \{(x_i,y_i) \}^m_{i=1} $ where each $x_i$ is a $d$-dimensional vector and each $y_i$ is a $p$-dimensional vector, meaning for each example of feature $x_i$ we get $p$ number of possible outputs. Thus $X$ is $m\times d $ dimensinal and $Y$ is $m \times p$ dimensional.
As we know $$Y=\phi(X) W$$
Assuming $\phi(X)$ remains same dimension as $X$, $W$ would be $d \times p$ dimensional.
For single output case, squared error loss function was
\begin{align}
    J(W) = & \sum_{i=1}^m || W^T x_i - y_i ||^2_2 \\
    J(W) = &  || X W - y ||^2_2 \\
    \bigtriangledown_W J= & \bigtriangledown_W \Big[ (XW-Y)^T (XW-Y) \Big]
\end{align}
Where W was $d \times 1$ dimensional.

But now for multiple outputs of $y_i$, assuming outputs are independet of prior, 

squared error loss function for each possible $p$ output will be :
\begin{align}
    J(W_p) = &  || X W_p - y_p ||^2_2 \\
     \bigtriangledown_{W_p} J= & \bigtriangledown_{W_p} \Big[ (XW-Y)^T (XW-Y) \Big] \\
                            = & X^TX W_p -X^T y_p = 0 \\
    thus, \, W_p = & (X^TX)^{-1} X^T y_p                        
\end{align}



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## Answer 4(b)

#####################

From the proof above we can say that if we independently solve $p$ classical linear regression an and get $W$ parameter $p$ times, we solve the multivariate regression task.

In this problem we assumed outputs are independent of prior but in baysian approximation where we have a prior on output, for multivate output we need to consider all the output prior for future $W$ parameter updates and thus can't calculate posterior output inpendently.

consider noisy conversation $y = f(x) + \epsilon = \phi(x)^T w + \epsilon$
according to bayes rules:

$ P_\phi(w|y,X) = \frac{ P_\phi(y|X,w)P(w) }{P_\phi(y|X) } $

$L = ||Y - XW||^2$


$W = (X^TX)^{-1}X^TY$

$L = ||Y - XW||^2 + ||XW - XW_R||^2$


```python

```

58fe274bb58617acd3c81e4bf52b14a940801b72
ipynb
Jupyter Notebook
assigment 1/problem 4.ipynb
SaminYeasar/COMP-652-Machine-Learning
510e0f237b780ec8465b8081487421171a6e5056
MIT
assigment 1/problem 4.ipynb
SaminYeasar/COMP-652-Machine-Learning
510e0f237b780ec8465b8081487421171a6e5056
MIT
assigment 1/problem 4.ipynb
SaminYeasar/COMP-652-Machine-Learning
510e0f237b780ec8465b8081487421171a6e5056
MIT
Qwen/Qwen-72B
1. YES
2. YES