Accuracy Maximization Analysis for Sensory-Perceptual Tasks: Computational Improvements, Filter Robustness, and Coding Advantages for Scaled Additive Noise
[['Perception science seeks to determine how perceiving organisms estimate behaviorally relevant properties of the environment based on proximal stimuli captured by the senses .', 'Understanding the details of the sensory-perceptual processing that support these abilities with natural stimuli is a primary focus of research .', 'It is widely appreciated that some stimulus features are more useful for some tasks than others , more likely to increase a given neuron’s response rate than others , and more likely to excite neurons in one brain area than another .', 'This specificity suggests that perceptual and neural performance in particular tasks is driven by sets of features that are of much lower dimensionality than the proximal stimuli themselves .', 'As a consequence , methods for reducing stimulus dimensionality are in widespread use in perception and neuroscience research .', 'Models of information encoding with natural stimuli are often developed without regard to what information will be decoded from the encoded signals .', 'Efficient coding , and many statistical methods for data characterization ( e . g . PCA , ICA ) , are designed to capture statistical properties of proximal ( observable ) stimuli without explicit consideration of the sensory-perceptual or behavioral goals for which the encoded information will be used [1 , 2] [3–9] .', 'The efficient coding hypothesis has been remarkably influential .', 'However , as Simoncelli & Olshausen ( 2001 ) point out , the hypothesis “states only that information must be represented efficiently; it does not say anything about what information should be represented” [7] .', 'Empirical studies in psychophysics and systems neuroscience often focus on the behavioral limits and neurophysiological underpinnings of performance in specific tasks [10–21] .', 'Thus , there is a partial disconnect between popular task-independent theories of encoding ( e . g . efficient coding ) and the methodological practices often followed by psychophysics and sensory and systems neuroscience .', 'Accuracy Maximization Analysis ( AMA ) provides a principled , data-driven approach to finding the stimulus features that are most useful for specific tasks ( e . g . estimation of a variable latent in the stimulus ) [22] .', 'AMA thus addresses a need that is not directly addressed by standard efficient encoding frameworks .', 'In conjunction with carefully calibrated natural image databases [21–26] , AMA has provided predictions for the encoding filters ( receptive fields ) that support optimal performance in several fundamental tasks in early vision [21–24 , 26] .', 'These receptive fields have , in turn , aided the development of ideal observers for the estimation of figure-ground , defocus blur , binocular disparity , retinal speed , and motion-in-depth [21 , 23 , 24 , 27] .', 'The predictions of these ideal observers are biologically plausible , dovetail with available neurophysiological data , and can tightly predict human performance with natural and artificial stimuli [21] .', 'These results may represent the beginnings of an important step forward in our ability , as a science , to develop ideal observer theories of mid-level visual tasks that act directly on natural retinal images .', 'AMA does not come without a set of constraints and disadvantages .', 'The most important constraint is that the stimuli must be contrast normalized before processing .', 'This constraint is appropriate for many perceptual tasks for which the task-relevant information is contained in the pattern of contrast over space and time , but it renders the method ill-suited for tasks in which the primary source of information is contained in the magnitude ( intensity ) of a stimulus .', 'Second , the AMA cost landscape is non-convex , so guarantees cannot be made that local minima found by the method represent the global minimum; standard techniques for protecting against non-global local minima must be used ( e . g . random starts ) .', 'However , for the set of problems for which AMA is well-suited , its most glaring disadvantage is its computational cost: compute time is quadratic in the number of elements in the training set .', 'Without specialized computing resources , the computational cost renders the method impractical for use on large-scale problems .', 'The aims of this paper are four-fold .', 'First , to set our contribution in context , we re-derive the original equations for AMA [22] , developing intuitions along the way .', 'Second , we derive the gradient of the cost ( objective function ) for two popular cost functions—0 , 1 cost ( L0 norm ) and squared error cost ( L2 norm ) —and implement a stochastic gradient descent procedure for filter learning , which we call AMA-SGD .', '( source code at: http://www . github . com/BurgeLab/AMA ) .', 'These advances significantly reduce the method’s compute time , thereby rendering it a more practical tool for research on problems of wide spread interest in vision research and sensory and systems neuroscience .', 'Third , we show that AMA can be used to examine the relative impact on optimal coding of stimulus variability and priors over the latent variable .', 'Fourth , we show how scaled additive encoding noise ( i . e . additive noise with response variance proportional to the response mean ) and correlated ( i . e . non-orthogonal ) filters can interact to confer coding advantages in certain tasks .', 'We believe that the work presented here may help establish a general normative framework for understanding the diversity of tuning and response properties exhibited by neurophysiological receptive fields in cortex , and how they may contribute to task-specific processing of sensory stimuli .'], ['Accuracy Maximization Analysis ( AMA ) provides a closed-form expression for the optimal ( nonlinear ) decoding rule given five factors:', 'i ) a well-defined task ( i . e . a latent variable to estimate from high-dimensional stimuli ) ,', 'ii ) a labeled training set of stimuli ,', 'iii ) a particular set of filters ( receptive fields ) ,', 'iv ) a noisy filter response model , and', 'v ) a cost function ( Fig 1A ) .', 'Given these factors , the problem of finding the encoding filters that are optimal for a particular task reduces to searching for the filters that minimize the cost ( Fig 1B ) .', 'The Background and Setup section is ordered to follow the block diagram in Fig 1A .', 'Gradient descent routines require the gradient of the cost function .', 'The gradient must be determined numerically ( e . g . finite differences ) if an analytic expression is not known .', 'The computational cost of numerically evaluating the gradient is proportional to the number of dimensions required to define the gradient itself .', 'Methods for numerical evaluation typically proceed by taking a small step of size ε in each of Nd directions .', 'In our case , Nd is the number of dimensions that define each filter ( i . e . receptive field ) .', 'For example , a filter that processes a 15x15 pixel image is defined by 225 dimensions , one dimension for each pixel .', 'Thus , the gradient of the cost with respect to the filter is 225 dimensional .', 'An analytical expression for the gradient can be expected to yield computational savings equal to a factor Nd less the time required to evaluate the gradient .', 'This improvement in speed can be substantial for problems in which the stimuli and filters are relatively high dimensional .', 'Here , we derive the gradient of the cost for two popular cost functions: the 0 , 1 cost function ( i . e . L0 norm ) and the squared error cost function ( i . e . L2 norm ) .', 'These two cost functions are commonly used in the fields of vision research , visual neuroscience , statistics , and machine learning .', 'They also represent opposite extremes of commonly used cost functions .', 'The 0 , 1 cost function penalizes all errors equally , regardless of their magnitude .', 'The squared error cost function penalizes small errors minimally and large errors severely .', 'We reason that if the behavior of the algorithm is understood for these two cost functions , reasonable inferences can be made about algorithm’s behavior for intermediate cost functions ( e . g . L1 norm ) .', 'The optimal estimator for the L0 norm cost function is the maximum a posteriori ( MAP ) estimator ( see S4 Text ) .', 'In the present case , the expected L0 cost across all stimuli is closely related to the Kullback-Leibler ( KL ) divergence between the observed posterior and an idealized posterior with all its mass at the correct level of the latent variable ( S5 Text ) ; for both statistics , the expected cost is a function only of the probability mass at the correct level of the latent variable .', 'Thus , the appropriate estimator for both measures is the posterior maximum ( i . e . the MAP estimator ) .', 'The optimal estimator for the L2 norm cost function is the mean of the posterior probability distribution , or the minimum mean squared error ( MMSE ) estimator ( see S6 Text ) .'], ['Binocular disparities are the local differences between the left and right eye retinal images due to the different vantage point each eye has on the world .', 'Binocular disparities are used for fixating the eyes and for computing the depth structure of scenes ( Fig 6A ) .', 'But the disparities themselves must be estimated before they can be used for depth perception .', 'The estimation of binocular disparity is a classic problem in vision science , and is often referred to as the stereo-correspondence problem .', 'The behavioral limits and neural mechanisms of disparity estimation have been extensively investigated [15 , 18 , 20 , 38–41] .', 'However , until recently there was no ideal observer for estimating disparity in natural images [24] .', 'To develop this ideal observer , Burge & Geisler ( 2014 ) first obtained a labeled training set of randomly selected 1 deg binocular retinal images of natural scenes with disparities ranging between -15 to 15 arcmin ( 400 binocular stimuli x 19 disparity levels = 7600 total ) .', 'Physiological optics , and the wavelength sensitivity and spatial sampling of the foveal photoreceptors were accurately modeled .', 'AMA was then used to find the small population ( n = 8 ) of binocular filters that extract the most useful information in natural images for the task .', 'Additional filters yielded little improvement , suggesting that eight binocular filters capture most of the available task-relevant information .', 'The properties of the filters mimic the receptive fields of disparity sensitive neurons in cortex , and optimal disparity decoding predicts many aspects of human disparity estimation and discrimination performance .', 'Please see Burge & Geisler ( 2014 ) for extensive details on the training set , the ideal observer for disparity estimation , and the role AMA played in its development .', 'The two most useful filters in the disparity estimation task are shown in Fig 6B .', 'These receptive fields took approximately 1 hour to learn on a 2012 MacBook Pro .', 'The disparity-conditioned filter responses p ( r|X ) to the contrast normalized stimuli are approximately Gaussian ( Fig 6C ) , and the optimal filters are somewhat anti-correlated: ρ=f1Tf2=−0 . 22 .', 'Posterior probability distributions for three joint filter responses ( oversized dots ) are shown in Fig 6D .', 'As the responses get farther from the origin , the posterior probability distributions have more of their mass at the correct level of the latent variable .', 'The filter response distributions in Fig 6C , and the manner in which they change with the value of the latent variable , are similar to the response distributions obtained for other tasks/coding problems that have been modeled with ‘energy-like’ computations ( e . g . disparity-energy , motion-energy ) [15 , 21 , 24 , 42]: the information about the latent variable is carried primarily by the covariance of the filter responses .', 'This characteristic pattern of filter response will inform subsequent analyses of how interactions between filter correlation , response noise , and stimulus distributions impact encoding fidelity ( see Discussion ) .', 'In this section , we examine the robustness of the optimal filters to changes in the prior probability distribution , overall noise power , and cost function .', 'We find that the optimal filters are remarkably stable , suggesting that natural stimulus properties are the primary determinants of the optimal filter shapes .'], ['Standard forms of the most popular methods for dimensionality reduction ( e . g . PCA ) and statistical characterization ( e . g . ICA ) do not include a specific model of encoding noise .', 'In such models , any set of receptive fields ( i . e . basis elements ) spanning the same subspace encode an arbitrary stimulus with equivalent fidelity .', 'In other words , the encoding provided by a given pair of filters within their spanned subspace is not unique .', 'This fact is due to an assumption common to a large class of popular methods for dimensionality reduction: namely , that the filters encode stimuli noiselessly .', 'Encoding noise corrupts measurements by real biological or machine vision systems .', 'AMA incorporates an explicit noise model at the level of the encoding filters ( Eqs 1a–1d ) , as do probabilistic extensions to PCA and ICA[50 , 58] .', 'Encoding noise ( i . e . the filter response model ) can make the stimulus encoding unique within the subspace that the encoding filters define .', 'Figs S1 , 13 and 14 are designed to help develop a geometric intuition for why filter response noise can make the encoding of particular filters within a subspace unique .', 'After building intuition , we discuss the implications of this fact for our understanding of neural coding .', 'We consider three classes of encoding filter response:', 'i ) a noiseless response model ( α = 0 and σ02=0 , see Eq 1d; S1 Fig )', 'ii ) an constant additive response model ( α = 0 and σ02>0; Fig 13 ) , and', 'iii ) a scaled additive response noise model ( α > 0 and σ02>0 ) ; Fig 14 ) .', 'We also consider the impact of having orthogonal ( i . e . uncorrelated ) encoding filters vs . non-orthogonal ( i . e . correlated or anti-correlated ) encoding filters .', 'We will see that the type of noise ( constant additive or scaled additive ) , filter correlation ( i . e . redundancy ) , and filter orientation in the subspace can interact non-trivially to confer coding advantages .', 'Consider two stimuli that are projected into a standard basis spanned by an arbitrary pair of filters f; let this subspace be represented by orthonormal basis e ( Figs S1 , 13 and 14 ) If noiseless encoding is assumed ( which is of course biologically unrealistic ) , the stimuli are encoded with equal fidelity no matter the filter correlation ( redundancy ) or rotation , so long as the filters lie in the same subspace .', 'Specifically , filters F1 and F2 encode the stimulus identically well , regardless of whether the encoding filters are positively correlated , orthogonal , or anti-correlated .', 'Rotating the encoding filters in the subspace also has no impact on coding fidelity .', 'Thus , with no encoding noise , every set of filters spanning the same subspace provides an equivalent stimulus encoding ( S1 Fig ) .', 'With constant additive response noise the situation changes .', 'Now , filters F1 and F2 encode the stimulus with different fidelity when they are correlated vs . when they are orthogonal; note the differences in the uncertainty ellipses ( Fig 13A–13C ) .', 'When the filters are orthogonal ( Fig 13B and 13E ) , the uncertainty ellipses are circular , and stimulus encoding remains invariant to rotation ( Fig 13E ) .', 'Stimulus encoding by correlated filters , however , is no longer invariant to filter rotation ( Fig 13A , 13D , 13C and 13F ) .', 'With scaled additive response noise , the situation changes still further .', 'Filters F1 and F2 now provide a unique encoding of the stimulus , regardless of whether they are correlated or uncorrelated , and regardless of whether the filters are rotated or not ( Fig 14A–14F ) .', 'The fact that the fidelity of stimulus encoding changes as a function of filter correlation and rotation within a subspace suggests that encoding cost ( i . e . the value of AMA objective function ) may depend on the particular filters within a given subspace .', 'To examine this issue quantitatively , we rotated the optimal receptive field pair within their spanned subspace and computed the cost for each rotation angle θ .', '( see S8 Text ) .', 'Recall that the optimal filters in the current task are somewhat anti-correlated ( ρ=f1Tf2=−0 . 22 ) .', 'We also examined the cost of forcing the filters to be orthogonal .', 'To do so , we performed Gram Schmidt orthogonalization , rotated the orthogonalized filters , and computed the cost for each rotation angle θ .', 'Fig 15 shows that some filter pairs within the subspace yield lower cost than others ( Fig 15A–15C ) .', 'Example filter pairs that have been rotated by different amounts are depicted in Fig 15D .', 'With scaled additive response noise ( the noise model with which the filters were learned ) , cost is lowest for the optimal filters .', 'For all non-zero rotation angles cost increases , except for 180° ( Fig 15A ) .', '( A 180° rotation angle corresponds to contrast reversal of both receptive fields , which by assumption ( Eqs 1a–1d ) , gives identical performance to the original filters ) .', 'If the filters are orthogonalized , cost increases on average .', 'More importantly , the minimum cost of the best pair of orthgonalized filters is higher than the minimum cost of the original somewhat anti-correlated filters .', 'This result shows that correlated filters can provide a coding advantage over orthogonal filters in the AMA framework .', 'Next , we examined constant additive response noise models .', 'With constant additive ( instead of scaled additive ) response noise , cost is also modulated by the rotation angle within the subspace , but only if the filters are non-orthogonal .', 'If the filters are orthogonal , all filter rotations within the subspace provide an identical encoding ( Fig 15A ) .', 'These results are consistent with the intuitions developed in Fig 13 .', 'To make a quantitative comparison between the encoding costs associated with the two noise models , we matched the noise power between the two models .', 'Specifically , we set the constant additive noise variance equal to the average variance of the scaled additive noise σconstant2=1N∑klσscaled2 where N is the total number of stimuli in the training set .', 'Encoding filters having this constant additive noise never achieve costs as low as the scaled additve noise model ( Fig 15A and 15C ) .', 'So long as noise power is matched , this result holds whether the filters are learned with scaled additive or constant additive noise .', 'Therefore , for the task considered here ( disparity estimation ) , scaled additive noise provides a coding advantage .', 'This same result holds for several other fundamental tasks in early vision with natural images ( retinal speed estimation , motion-in-depth estimation ) .', 'These tasks have all been successfully modeled with energy-like computations ( disparity energy model , motion energy model , etc . ) .', 'We conclude that scaled additive noise provides a coding advantage over constant additive noise in an important class of estimation tasks in early- and mid-level vision for which energy-like computations are appropriate .', 'There are several take-away points .', 'First , in AMA , all encoding filters , even those spanning the same subspace , do not provide equivalent encodings .', 'Second , correlated filters can yield lower cost encodings than orthogonal filters .', 'Third , scaled additive response noise can yield lower cost encodings than constant additive response noise when the noise power ( i . e . average noise variance ) is matched .', 'These results have implications for how to think about the pros and cons of the constraints imposed on many methods for dimensionality reduction .', 'In this section , we examine why scaled additive response noise can provide an advantage over constant additive response noise .', 'In early visual areas , neural response variance increases approximately linearly with the mean response [31 , 32] .', 'Much attention has been paid to this property of neural response , especially as it relates psychophysical performance in target detection , a paradigmatic task in the spatial vision literature .', 'When response variance is proportional to the mean response , a single neuron’s signal-to-noise ratio for detection of a particular target is proportional to the square-root of the mean response , SNR∝r .', 'On the other hand , if neural response variance is independent of the mean response ( i . e . constant ) , the signal to noise ratio is proportional to the mean response , SNR ∝ r .', 'Thus , it has been sensibly argued that , all other things equal , scaled additive noise must have deleterious effects on neural coding compared to constant additive noise .', 'However , in the case considered above , scaled additive noise ( response noise variance proportional to mean response ) outperforms constant additive noise with matched noise powers .', 'Many ( most ? ) visual tasks are performed at super threshold contrasts and involve estimating the value of a variable that is latent in the proximal stimulus .', 'In latent variable estimation and discrimination tasks , scaled additive noise can benefit , rather than deteriorate the quality of neural encoding .', 'This is true in the current task of disparity estimation ( Figs 6 and 14–16 ) .', 'It is also true of other related tasks in early vision ( e . g . speed estimation and motion-in-depth estimation ) .', 'This result raises the possibility that a ubiquitous neural response property that hurts performance in contrast detection tasks may actually benefit performance in tasks that are somewhat ‘higher-level’ ( e . g . disparity estimation ) .', 'Why , in latent variable estimation and discrimination , can a vision system with scaled additive noise outperform a vision system with matched constant additive noise ?', 'Some development is necessary to answer this question because the answer depends on a set of interlocking dependencies between the noise model , filter correlations , and the latent-variable-conditioned stimulus distributions p ( s|Xi ) .', 'When the task is to discriminate one latent variable value from another ( as opposed to detecting a well-defined contrast pattern—a signal-known-exactly task ) , it is less clear what constitutes ‘signal’ and what constitutes ‘noise’ .', 'We have found it useful to approach the problem with standard techniques in the pattern classification literature [59] .', 'Consider a hypothetical case that illustrates the relevant principles .', 'Fig 16A–16C shows three simulated stimulus distributions projected into the subspace spanned by a pair of filters .', '( These simulated distributions are superficially similar to the disparity conditioned stimulus distributions shown in Fig 6C . )', 'These same exact stimuli are encoded by three pairs of filters that are differently correlated , but that have the same scaled additive noise and that span the same subspace ( Eqs 1a–1c ) .', 'This subspace is represented by the orthonormal basis e that spans the same subspace as the filters f .', 'The upper and lower rows of Fig 16A–16C represent the same information in different forms .', 'In the upper row , three latent-variable-conditioned stimulus distributions are projected into the subspace defined by a pair of filters p ( eTs|Xi ) ; the dots represent the stimulus projections in the standard basis and the ellipses represent encoding uncertainty .', 'In the lower row , the exact same stimulus projections are represented by the mean responses that they elicit from each filter pair , p ( fTs|Xi ) ; the dots represent the stimulus projections onto the filters and the ellipses represent filter response noise .', 'We refer to the lower row as the filter basis .', 'Consistent with the assumption that the filter response noise is independent ( Eqs 1c and 1d ) , all the noise ellipses in the filter basis ( lower row ) are aligned with the axes of the space ( i . e . the noise covariance matrix is diagonal ) .', 'The linear mappings from the filter basis to the standard basis and back are derived in S8 Text .', 'The oblique orientations of the uncertainty ellipses in the upper row of 16ac therefore do not reflect noise correlations [60 , 61] .', 'Now , examine the effect of changing the filters from being positively correlated ( Fig 16A ) , to orthogonal ( Fig 16B ) , to anti-correlated ( Fig 16C ) .', 'As filter correlation and orientation within the subspace changes ( see Fig 16D ) , the uncertainty ellipses ( upper row ) change their orientation .', 'Cost is minimized when the uncertainty ellipses maximally align with the projections of the conditional stimulus distributions ( Fig 16E ) .', 'Remarkably , filter correlation ( i . e . cosine similarity ) , filter orientation in the subspace , and scaled additive noise can conspire to align the uncertainty ellipses with the conditional stimulus distributions .', 'The conditional distributions of filter responses are shown in the lower row of Fig 16A–16C .', 'The mean filter response to each stimulus is obtained by projecting the stimulus onto each filter ( c . f . Fig 16D; r = fTs ) .', 'In filter response space ( i . e . the filter basis ) , two effects occur as filter correlation changes .', 'The most dramatic effect is the change in the distribution the response means .', 'A secondary effect is that the height and/or width of the response noise ellipses decrease as the corresponding mean filter response approaches zero .', 'Note , however , that the noise ellipses always remain aligned with the cardinal axes .', 'In other words , the noise ellipses have diagonal covariance matrices , consistent with the assumption of independent response noise .', 'In filter response space , the algorithm’s aim is to position the filters such that the conditional response distributions p ( R|Xi ) are as discriminable from each other as possible .', 'With a constant additive noise model , in the standard basis , all uncertainty ellipses have the same orientation ( c . f . Fig 13B ) ; in the filter basis , all noise ellipses are circular ( i . e . equal variance diagonal covariance matrices ) .', 'Thus , if the conditional stimulus distributions change orientation as a function of the value in the latent variable ( as they do here ) , the constant additive noise model cannot align the uncertainty ellipses with the stimulus distributions across the space .', 'As a consequence , encoding cost increases ( Fig 15A , S2 Fig ) .', 'In general , cost is minimized when encoding uncertainty is maximized within , and minimized between , latent-variable-conditioned stimulus distributions .', 'That is , when uncertainty due to noise maximally overlaps the uncertainty due to ‘nuisance’ stimulus variation , coding of the latent variable is improved .', 'A related claim about the potential utility of noise correlations has recently been made [61 , 62] .', 'Additionally , scaled additive noise yields lower response variance for stimuli near the origin than stimuli far from the origin of the response space , thereby reducing the relative cost of ‘hard’ stimuli and increasing the relative cost of ‘easy’ stimuli .', 'Each stimulus therefore contributes more evenly to the cost .', 'In the tasks considered , this property causes the algorithm to make better use of the information provided by each stimulus .', 'If the expressions underlying AMA were reformulated as a learning rule , we suspect that scaled additive noise would enable the system to learn more efficiently .', 'Most importantly , we have shown that scaled additive noise and non-orthogonal filters can confer significant benefits to neural encoding .', 'These benefits are obtained when uncertainty due to noise is shaped to match within-level stimulus variation .', 'The AMA cost landscape is non-convex , so there is no guarantee that the filters found by the algorithm indeed represent the global minimum .', 'However , there are several reasons to suspect that the filters for disparity estimation presented here indeed found a global minimum .', 'First , the recovered filters occupy the minimum of the cost landscape within the subspace that they span ( Fig 15A–15C ) .', 'Second , somewhat surprisingly , correlated ( non-orthogonal ) filters with scaled additive noise tend yield lower cost landscapes with deeper minima ( Fig 15B and 15C ) than orthogonal filters with constant additive noise .', 'Third , the work presented here and in previous publications has found that different random initializations tend to yield equivalent filters .', 'The response model used here allows both positive and negative encoding filter responses whereas real neurons give only positive responses .', 'Future work will examine the pros and cons of incorporating half-rectification into the response model ( Eqs 1a–1d ) .', 'One drawback of incorporating half-rectification is that more filters will be required to cover the same response space , thereby increasing the dimensionality of the search space , perhaps leading to less stable performance .', 'However , incorporating half-rectification will increase biological realism , allow for differential sensitivity to ON/OFF contrast changes [63] , and increase the flexibility of the system to match stimulus encoding uncertainty to task-irrelevant stimulus variation ( Figs 13–16 ) .', 'Interest in neural systems identification has surged in recent years .', 'The field has generated a slew of models with ever increasing sophistication and descriptive power .', 'Many of these models are known as ‘subunit models’ .', 'Subunit models seek to provide a computational level description of a neuron’s processing that can predict a neuron’s response to arbitrary stimuli .', 'The spike-triggered average ( STA ) and spike-triggered covariance ( STC ) analysis are early examples of subunit models [64–67] .', 'The generalized linear model ( GLM ) and generalized quadratic model ( GQM ) are examples of more recently developed subunit models that are more flexible and powerful [68–70] .', '( All of these methods have been adapted to handle non-spiking , real-valued data ( e . g . response rates or intracellular voltages[69 , 71] ) .', 'These methods have been widely adopted by the neuroscience community because of their success in providing compact , interpretable characterizations of the input-output relationship between stimuli and neural response .', 'In general , subunit models describe neural response with a low-dimensional set of stimulus features ( i . e . subunit receptive fields ) , a nonlinear pooling rule , a static output non-linearity , and noise function that generates output noise .', 'As these models have increased in descriptive power and mathematical elegance , interest has increased in whether the computational components can be mapped back to specific biophysical components .', 'For example , in a subunit model description of a complex cell , one may ask whether presynaptic simple cells are the biophysical analogs of the model subunits .', 'A limitation of this class of subunit models is that although they can recover the subspace spanned by a set of receptive fields , the models cannot recover the subunit receptive fields themselves .', 'In traditional subunit models , any set of receptive fields spanning the same subspace encodes a given stimulus with equal fidelity .', 'This property of subunit models is due to the fact that they implicitly assume noiseless encoding .', 'AMA , on the other hand , has an explicit model of response noise for each filter ( i . e . subunit receptive field ) .', 'As discussed above , response yields encodings that are unique within the subspace defined by the filters ( Figs 15 , 16 and S1 ) .', 'By adapting AMA as a method for neural system’s identification , we speculate that it may be possible to identify both the subspace spanned by the subunit receptive fields , and the individual subunit receptive fields themselves .', 'As neural datasets come online having simultaneous recordings between ‘target cells’ and their presynaptic inputs ( e . g . connected V1 and LGN units ) ,
['Introduction', 'Methods', 'Results', 'Discussion']
['Accuracy Maximization Analysis ( AMA ) is a recently developed Bayesian ideal observer method for task-specific dimensionality reduction .', 'Given a training set of proximal stimuli ( e . g . retinal images ) , a response noise model , and a cost function , AMA returns the filters ( i . e . receptive fields ) that extract the most useful stimulus features for estimating a user-specified latent variable from those stimuli .', 'Here , we first contribute two technical advances that significantly reduce AMA’s compute time: we derive gradients of cost functions for which two popular estimators are appropriate , and we implement a stochastic gradient descent ( AMA-SGD ) routine for filter learning .', 'Next , we show how the method can be used to simultaneously probe the impact on neural encoding of natural stimulus variability , the prior over the latent variable , noise power , and the choice of cost function .', 'Then , we examine the geometry of AMA’s unique combination of properties that distinguish it from better-known statistical methods .', 'Using binocular disparity estimation as a concrete test case , we develop insights that have general implications for understanding neural encoding and decoding in a broad class of fundamental sensory-perceptual tasks connected to the energy model .', 'Specifically , we find that non-orthogonal ( partially redundant ) filters with scaled additive noise tend to outperform orthogonal filters with constant additive noise; non-orthogonal filters and scaled additive noise can interact to sculpt noise-induced stimulus encoding uncertainty to match task-irrelevant stimulus variability .', 'Thus , we show that some properties of neural response thought to be biophysical nuisances can confer coding advantages to neural systems .', 'Finally , we speculate that , if repurposed for the problem of neural systems identification , AMA may be able to overcome a fundamental limitation of standard subunit model estimation .', 'As natural stimuli become more widely used in the study of psychophysical and neurophysiological performance , we expect that task-specific methods for feature learning like AMA will become increasingly important .']
['In psychophysics and neurophysiology , the stimulus features that are manipulated in experiments are often selected based on intuition , trial-and-error , and historical precedence .', 'Accuracy Maximization Analysis ( AMA ) is a Bayesian ideal observer method for determining the task-relevant features ( i . e . filters ) from natural stimuli that nervous systems should select for .', 'In other words , AMA is a method for finding optimal receptive fields for specific tasks .', 'Early results suggest that this method has the potential to be of fundamental importance to neuroscience and perception science .', 'First , we develop AMA-SGD , a new version of AMA that significantly reduces filter-learning time , and use it to learn optimal filters for the classic task of binocular disparity estimation .', 'Then , we find that measureable , task-relevant properties of natural stimuli are the most important determinants of the optimal filters; changes to the prior , cost function , and internal noise have little effect on the filters .', 'Last , we demonstrate that some ubiquitous properties of neural systems , generally thought to be biophysical nuisances , can actually improve the fidelity of neural codes .', 'In particular , we show for the first time that scaled additive noise and redundant ( non-orthogonal ) filters can interact to sculpt uncertainty due to internal noise to match task-irrelevant natural stimulus variability .']
['learning', 'tangents', 'ellipses', 'social sciences', 'geometry', 'learning and memory', 'neuroscience', 'probability distribution', 'mathematics', 'cognitive psychology', 'computational neuroscience', 'vision', 'coding mechanisms', 'animal cells', 'probability theory', 'psychology', 'cellular neuroscience', 'cell biology', 'neurons', 'biology and life sciences', 'cellular types', 'physical sciences', 'computational biology', 'sensory perception', 'cognitive science']