Title: Nonlinear dynamics of localization in neural receptive fields
Venue: NeurIPS 2024
Accepted: yes
Keywords: localization

receptive fields

learning dynamics

emergence
Abstract: Localized receptive fields—neurons that are selective for certain contiguous spatiotemporal features of their input—populate early sensory regions of the mammalian brain. Unsupervised learning algorithms that optimize explicit sparsity or independence criteria replicate features of these localized receptive fields, but fail to explain directly how localization arises through learning without efficient coding, as occurs in early layers of deep neural networks and might occur in early sensory regions of biological systems. We consider an alternative model in which localized receptive fields emerge without explicit top-down efficiency constraints—a feed-forward neural network trained on a data model inspired by the structure of natural images. Previous work identified the importance of non-Gaussian statistics to localization in this setting but left open questions about the mechanisms driving dynamical emergence. We address these questions by deriving the effective learning dynamics for a single nonlinear neuron, making precise how higher-order statistical properties of the input data drive emergent localization, and we demonstrate that the predictions of these effective dynamics extend to the many-neuron setting. Our analysis provides an alternative explanation for the ubiquity of localization as resulting from the nonlinear dynamics of learning in neural circuits

Nonlinear dynamics of localization in neural receptive fields

Abstract

Localized receptive fields-neurons that are selective for certain contiguous spatiotemporal features of their input-populate early sensory regions of the mammalian brain. Unsupervised learning algorithms that optimize explicit sparsity or independence criteria replicate features of these localized receptive fields, but fail to explain directly how localization arises through learning without efficient coding, as occurs in early layers of deep neural networks and might occur in early sensory regions of biological systems. We consider an alternative model in which localized receptive fields emerge without explicit top-down efficiency constraints-a feedforward neural network trained on a data model inspired by the structure of natural images. Previous work identified the importance of non-Gaussian statistics to localization in this setting but left open questions about the mechanisms driving dynamical emergence. We address these questions by deriving the effective learning dynamics for a single nonlinear neuron, making precise how higher-order statistical properties of the input data drive emergent localization, and we demonstrate that the predictions of these effective dynamics extend to the many-neuron setting. Our analysis provides an alternative explanation for the ubiquity of localization as resulting from the nonlinear dynamics of learning in neural circuits. 1

1 Introduction

A striking feature of peripheral responses in the animal nervous system is localization -that is, the linear receptive fields of simple-cell neurons often respond to contiguous regions much smaller than their full input domain. In vision, retinal ganglion cells approximate localized center-surround filters that tile the input [Dac+00; Doi+12; KK78], and simple cells downstream in primary visual cortex have localized filters that are selective for spatial frequency and orientation [HW59; HW68; RT95; NS08; WMG11; RSH02; Rin02]. In primary somatosensory cortex, neurons respond to stimulation of restricted regions of skin [Cro+11] and in primary auditory cortex, spatiotemporal receptive fields are typically localized in both time and frequency domains [DWZ03; HDZ08]; see Fig. 1 (left).

By contrast, artificial learning systems do not always learn localized filters. Principal component analysis tends to fit weights that span the entire input signal, as do unregularized autoencoder neural network architectures and restricted Boltzmann machines [Sax+11]. This difference has prompted the search for artificial learning models that can learn localized receptive fields from naturalistic stimuli, the most notable of which are sparse coding [OF96; OF97] and independent component analysis [ICA; BS97; vHvdS98]. Sparse coding, ICA, and related compression methods that produce localized receptive fields from naturalistic data share a top-down approach-they find an efficient representation of the input signal by optimizing an explicit sparsity criterion, or an independence criterion that necessitates sparsity in a critically parameterized regime [Fie99; Sax+11].

[Rin02]

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Though sparsity is appealing as a potentially unifying explanation for localization, localization also emerges naturally in networks trained to perform classification tasks without any explicit sparsity regularization [KSH12; ZF13; Yos+15; Sen+18]; see Fig. 1 (center) for an example. Ingrosso and Goldt [IG22] distilled such examples of emergent localization by demonstrating that localized receptive fields emerge in simple feedforward neural networks trained on a data model with properties meant to approximate natural visual input, in particular, locality structure (statistical independence of non-collocated dimensions) and non-Gaussianity (higher-order cumulants are non-null). In simulations, Ingrosso and Goldt [IG22] tie the dynamical emergence of localization to increased tuning to higher-order statistics of the input, and demonstrate that even a single neuron is sufficient to learn a localized receptive field in this setting.

In this work, we build on the demonstration of Ingrosso and Goldt [IG22] with the aim of describing the mechanisms behind the emergence of a localized receptive field in this minimal setting. The higher-order input statistics that drive localization are challenging to analyze with existing tools that exploit implied Gaussianity [Gol+20]. By separating two stages of learning, we are able to derive equations for the effective early-time learning dynamics of the single neuron model that learns a localized receptive field from idealized naturalistic data. Our analytical model identifies a concise description of the higher-order statistics that drive emergence, and we validate both positive and negative predictions of this analytical model via simulations with many neurons; see Fig. 1 (right). These findings suggest an alternative path to account for the ubiquity of localization in early neural responses as resulting from the interaction of the nonlinear dynamics of learning in neural circuits and naturalistic data with higher-order statitistical structure, rather than an explicit efficiency criterion.

2 Modeling approach

We extend the setting of Ingrosso and Goldt [IG22], a minimal example of a neural network that learns localized receptive fields from idealized naturalistic data. We analyze the dynamics of learning in this setting in Section 3 and validate our analytical model with simulations in Section 4.

2.1 Neural network architecture and learning algorithm

We consider a two-layer feedforward neural network with nonlinear activation and scalar output. While simple, this architecture is highly expressive, capable of approximating arbitrary integrable univariate functions with appropriate scaling [Bar93; Pin99], and exhibits rich feature learning dynamics that underlie the performance of models at scale [Woo+20], making this architecture the ongoing subject of theoretical neural network analyses [MMN18; Gol+19; Vei+22]. We denote a two-layer network with N -dimensional input, M hidden units, and one-dimensional scalar output as

where σ : R → R is a pointwise nonlinearity such as the rectified linear unit (ReLU) or sigmoid function, w (1) m ∈ R N and w (2) m ∈ R are learnable weights, b (1) m , b (2) ∈ R are learnable bias terms, and ⟨· , ·⟩ denotes the standard Euclidean inner (dot) product on R N . When the second-layer parameters are fixed, this model is known as a soft-committee machine [SCM; SS95], which [IG22] notes learns less noisy receptive fields but exhibits similar localization behavior. The many-neuron architecture in M1 is the focus of our simulations (Section 4), but the dynamics of this model are too complex to analyze directly, even for the idealized naturalistic data model considered here. In order to derive analytical results (Section 3), we consider the simplest neural network exhibiting the desired localization phenomenon, a single hidden neuron without bias and with rectified linear unit activation, written as

where ReLU( x ) = max( x, 0) , applied pointwise to vectorial input. As Ingrosso and Goldt [IG22] demonstrate, the localized receptive fields learned by the many- and single-neuron models defined in M1 and M2 are qualitatively similar up to spatial translation, which permits us to generalize insights from analyzing the learning dynamics of the single-neuron M2 to the many-neuron M1. For simulations, we initialize the weights and biases as independent draws from an isotropic Gaussian distribution with scaled variance, and train with batch gradient descent with a fixed learning rate on the mean-squared error (MSE) evaluated on input-output pairs from the task; see Section 2.3 for task sampling procedures.

2.2 Stimulus properties

The data model of Ingrosso and Goldt [IG22] can be shown to satisfy three conditions that enable the analysis we give in Section 3. We consider several other data models that share the below properties but differ in generative mechanism in order to probe the effect of these properties on localization. In particular, we consider data X sampled from distributions p on R N satisfying the following:

Stimulus properties 1-3 ( idealization of natural images ).

Properties S1 and S2 are defining characteristics of natural image data [HHH09]. Property S3 can also be seen to hold for natural images after centering and is convenient analytically because it implies that E [ X ] = 0 . Property S1 assumes that p is implicitly parameterized by N in order to state that the statistical dependence between entries of X vanishes as their separation increases. 2

We denote the covariance of X by Σ ≜ Cov[ X ] , the square of principal-diagonal entries (the variance of each entry of X ) by σ 2 , and the i -th row by σ i . Weak dependence (S1) implies that entries far from the principal diagonal of Σ will be 0, while translation-invariance (S2) implies that Σ is circulant ( i.e. , entries along each diagonal are equal) and thus identifiable by a single row; see Fig. 2 (center).

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2.3 Lengthscale discrimination task

Ingrosso and Goldt [IG22] develop a minimal task for which localization emerges in a feedforward neural network: binary discrimination between inputs from two distributions that differ in the lengthscale of the correlations between their entries. This lengthscale discrimination task can be seen as a pretext task for self-supervised learning [KZB19; Che+20] of representations [ cf. unsupervised: OF96; BS97]. More precisely, we generate data ( X , Y ) for supervised training according to

where p is to be defined, Σ y are distinct covariance matrices for each y , and we sample Y uniformly among a set of increasing lengthscale correlation classes y ∈ { 0 , 1 , . . . } , which correspond to the strength of correlation between distant positions. For instance, in the case of two classes ( y = 0 , 1 ), we take Σ 0 to be closer to σ 2 I N than Σ 1 , where I N is the N × N identity matrix and σ is a fixed value. This construction isolates, via distinct covariance matrices per class, the second-order statistics, which we will see below enter into the learning dynamics separately from other properties of p ( X ) , including, most critically, the implied marginal distributions, p ( X i ) .

Ising . The first distribution we consider is the one-dimensional Ising model. It is of interest as a distribution that satisfies S1 to S3 with marginals p ( X i ) with extreme support on {± 1 } , making it the distribution that promotes localization most strongly, as we will see in Section 3. In the absence of an external field, the Ising distribution is

where J is a chosen pairwise interaction strength, Z is the normalizing constant, and we enforce a periodic boundary constraint via x N +1 ≡ x 1 . As J increases, the lengthscale of the correlations in X also increases. For simulations, we sample from p Ising using a Gibbs sampler [GG84]. Discrimination tasks in the simulations in Section 4 use J 1 = 0 . 7 (for y = 1 ) and J 0 = 0 . 3 (for y = 0 ).

NLGP ( g ) . We also consider the data model used in Ingrosso and Goldt [IG22], the nonlinear Gaussian process (NLGP), which enables one to interpolate between distributions that do and do not yield localization via a single parameter, g . A sample X | Y = y from the NLGP is constructed by first sampling a Gaussian Z | Y = y ∼ N (0 , ˜ Σ y ) and then transforming it via

where erf is the Gauss error function, Z is a normalization constant to ensure that the variances of X i and Z i are the same, and ˜ Σ y is a covariance matrix for Z , where we use ( ˜ Σ y ) ij = exp( -( i -j ) 2 /ξ 2 ) for a lengthscale parameter ξ [IG22]. If g ≈ 0 (where localization is not observed), gZ i will tend to lie in the linear regime of erf , so Z i will be untransformed, i.e. , X is Gaussian. However, as g →∞ (where localization is observed), gZ i will tend to saturate erf , so X i will have support on {± 1 } .

Kur ( k ) . The final family we consider is chosen to give us flexibility over the kurtosis κ of the marginals p ( X i ) . In the Ising model, the excess kurtosis ( κ -3 ) of the marginals is fixed at -2 , while in NLGP ( g ) , it varies from -2 to 0 . This family allows us to vary the excess kurtosis from negative through positive values. We sample X | Y = y from this family via inverse transform sampling to vary the marginals while enforcing dependence via Gaussian copulas. More concretely, we sample Z | Y = y ∼ N (0 , ˜ Σ y ) and then transform it via

where ˜ σ is the standard deviation of Z i , Φ is the standard Gaussian cumulative distribution function (CDF), f is the CDF of the desired marginal distribution for X i , and Z is a normalization constant, which we compute numerically. We define ˜ Σ y as for NLGP . We choose f to be the generalized algebraic sigmoid function (see Appendix A.2) for k > 0 to make use of its tractable inverse, simplifying the procedure in Eq. (4). We denote the corresponding distribution by Kur ( k ) . Though we are able to continuously vary excess kurtosis, we lack an explicit form; however, numerical computation shows that for k ⪅ 5 . 8 , excess kurtosis is positive, while for k ⪆ 5 . 9 , it is negative.

3 Theoretical results

We derive an analytical model for the localization dynamics of the single-neuron architecture in M2. This result establishes necessary and sufficient conditions for localization under Properties S1 to S3 for the minimal case of a binary response, i.e. , y = 0 , 1 . The conditions for localization in the singleneuron architecture in M2 are demonstrated in Section 4 to also hold empirically for the many-neuron architecture in M1. Further, we use this model to derive a negative prediction about localization, that the architectures in M1 and M2 fail to learn a localized receptive field on elliptical distributions despite their non-Gaussian-in particular, significantly positive kurtosis-statistics [ cf. positive kurtosis as an objective or diagnostic for localization, HO00; IG22].

3.1 An analytical model for the dynamics of localization in a single neuron

Previous approaches to obtain analytical dynamics in the architectures in M1 and M2 have studied the gradient flow under the assumption that the preactivation ⟨ w , X ⟩ is approximately Gaussian [Gol+20; Ger+20; Gol+22], but this assumption fails to capture the propagation of higher-order statistics through a neural network that promotes localization [IG22]. Happily, the idealized visual input setting set out in S1 to S3 permits us some simplification. In particular, the translation-invariance of the data X under S2 and the architecture of M2 allow us to work with the marginal distributions of each input dimension, X i rather than the full joint distribution of X .

We now give the analytical simplifications that allow us to derive an analytical model for the localization dynamics of the single neuron architecture in M2, namely two assumptions on X | X i for all i ∈ { 1 , . . . , N } as a well as a mild condition on the weights that is satisfied at initialization. These are, where σ y i to denotes the i -th row of Σ y :

Analytical simplifications 1-3 ( early-time, limiting dynamics ).

(A1) E [ X | X i = x i , Y = y ] = x i σ y i , i.e. , the conditional mean scales linearly with x i .

(A2) Cov [ X | X i = x i , Y = y ] = Σ y -σ y i σ y ⊤ i , i.e. , the conditional covariance is smaller near i , but independent of the exact value of x i .

(A3) Lindeberg's condition holds for the sequence w 1 X 1 , . . . , w N X N | X i = x i as N →∞ for all x i .

Our motivation for Assumptions A1 to A3 is that they replicate the kurtosis of the marginal distributions X i (discussed further below) of two important and distinct limiting cases where localization does and does not appear, respectively: when X has support on the vertices of the hypercube {± 1 } N (satisfied by Ising for any J ), and when X is Gaussian (satisfied by NLGP with g ≈ 0 ).

The gradient flow in Lemma 3.1 also relies on Assumption A3 that Lindeberg's condition holds for the sequence w i X i , which ensures that no single term w i X i in the sequence can dominate. If this holds,

then we can conclude that ⟨ w , X ⟩ | X i is approximately Gaussian. As we discuss in Appendix C.2, this is almost always satisfied for a Gaussian initialization of w , and for slight deviations therefrom, and is satisfied by the settings of Ingrosso and Goldt [IG22]. Using this fact, we obtain an explicit form for the gradient flow early in training, stated in Lemma 3.1.

Lemma 3.1. Under Assumptions A1 and A2, the gradient flow for the single ReLU neuron in M2 early in training with y = 0 , 1 trained using MSE loss is

where o N (1) vanishes as N →∞ , and where φ : ( -1 , 1) → R is defined as

and alg -1 ( x ) = x/ √ 1 -x 2 , the inverse of the algebraic sigmoid function alg( x ) = x/ √ 1 + x 2 .

Lemma 3.1 reduces the study of higher-order statistics to the marginal distributions, X 1 , where, by translation invariance, all marginals have the same distribution, so we refer to X 1 without loss of generality. While Lemma 3.1 technically only holds early in training and breaks down if w becomes localized due to violation of A3, the gradient flow in Eq. (5) holds sufficiently long to detect the emergence of localization in the weights w . In particular, numerically integrating Eq. (5) yields localized weights w as t → ∞ . Moreover, the location of the peak of final weights from Eq. (5) corresponds closely to the actual peak of the weight, when we observe localization; see Section 4.2 for empirical validation of this fact. The primary difference observed is that the localized bump from Eq. (5) is less peaked than when computed exactly; see Fig. 3 for a comparison between experimentally observed localized receptive fields and theoretical predictions.

3.2 Necessary and sufficient conditions for emergent localization

To establish an exact threshold at which localization emerges requires solving Eq. (5), which is not possible exactly for general nonlinear differential equations. 3 Nevertheless, the form of Eq. (5) reveals that localization is driven solely by the first term. Indeed, the second term depends only on the second-order statistics of the data, and so can be held fixed as X is varied from a distribution that induces localization to one that does not. Secondly, one can see that the first term in Eq. (5) does not change as w is scaled, in contrast to the second term. As such, the second term in Eq. (5) serves to constrain the scale of w , distinct from localization, while the first is primarily concerned with the shape of w , and thus localization. This further motivates the first term, and thus φ , which we will refer to as the amplifier and which itself depends on properties of the data distribution p ( X ) , as a focus of study for understanding localization.

We present an analysis of φ in Appendix B.1 that reveals the role of the marginal distribution of the data in driving localization. For each marginal, φ ( a ) ≈ ( √ 2 /π ) a for a ≈ 0 . For larger a , φ depends more strongly on the data distribution and can be super-linear (sub-linear), i.e. , greater (smaller) than ( √ 2 /π ) a . Super-linear φ encourage entries in w that are large in some neighborhood to grow faster than those that are smaller, yielding localization. Linear and sub-linear φ are the opposite, encouraging oscillatory or flat weights by suppressing neighborhoods in w . However, super- and sub-linearity may not hold uniformly, as φ can be both over its domain (see Fig. 3, bottom row, black line). As an approximation, we consider a third-order Taylor expansion (red lines in Fig. 3, second column), which reveals that for the canonical setting of σ 2 = 1 , negative excess kurtosis of the marginals yields super-linearity , while positive excess kurtosis yields sub-linearity ; see Appendix B.1. This leads us to the following claim, which is validated by our simulations in Section 4:

Claim 3.2. For sufficiently large N , if the data X ∈ R N satisfies conditions S1 to S3 and has marginal distributions with sufficiently negative excess kurtosis, then Model M2 will learn localized receptive fields. Conversely, if the excess kurtosis is sufficiently positive , it will not.

As a minimal positive example, the distribution with the most negative excess kurtosis is the symmetric Bernoulli, with a value of -2 . In our setting, this corresponds to a data vector X with support on

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the vertices of the hypercube, {± 1 } N . As mentioned above, it can be seen from the law of total covariance combined with sign-symmetry that A1 and A2 hold exactly. Note that φ is the same for all such distributions, which leads us to Claim 3.2 that any distribution satisfying conditions S1 to S3 whose marginals are maximally concentrated will induce a localized receptive field in M2. Importantly, this claim includes the limiting case of Ingrosso and Goldt [IG22], g → ∞ in NLGP . It also includes the Ising model as another example, corroborating an observation for restricted Boltzmann machines [Har+20] that Ising data induces localization in a learning model. These claims are validated for the single-neuron model in Fig. 3 and in Section 4 for the many-neuron model.

3.3 Case study: Elliptical distributions fail to produce localization

Above, we assume weak dependence (S1) as it enables a focus on how the marginals control localization. As a first investigation into departures from this regime, we consider data X sampled from an elliptical distribution, where weak dependence may not hold. We specialize the definition of an elliptical distribution [Fra04] to our setting of multiple class labels and sign-symmetry:

Def 1. Samples X ∈ R N satisfy an elliptical distribution if we can write X | Y = y ( d ) = R y Λ y U y where R y is a nonnegative random variable, Λ y ∈ R N × D is such that Λ y Λ ⊤ y = Σ y , and U y is independent of R y and uniformly distributed on the D -dimensional sphere.

The class of elliptical distributions is broad, imposing only the constraint that the contours of the density be ellipses; the multivariate Gaussian and Studentt distributions are examples. As such, they can vary greatly in measures of non-Gaussianity, including kurtosis, while maintaining enough structure for analytical convenience. Proposition 3.3 states that training on elliptical data prevents localization in the single ReLU neuron model.

Proposition 3.3. Assume the data X are sign-symmetric (S3), translation-invariant (S2), and follow an elliptical distribution such that the MSE on the task in Section 2.3 is always finite. If Σ 0 , Σ 1 are such that the ratio of their i -th eigenvalues, λ i (Σ 0 ) /λ i (Σ 1 ) , assumes a particular value for at most two distinct i , then the steady states of the weight of M2 are sinusoids, i.e., not localized.

The condition on the number i such that the ratio of the i -th eigenvalues are the same constrains the number of Fourier components that can be non-zero in the steady state of w . While opaque, this requirement seems to always hold in practice, as even slight increases in length-scale correlation can dramatically change the spectrum of Σ y .

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The proposition is surprising because it reveals that the kurtosis of the preactivation is not an appropriate metric for explaining localization. Consider the example of the N -dimensional Studentt distribution with ν degrees of freedom, t N ( ν ) . If X ∼ t N ( ν ) , then ⟨ w, X ⟩ ∼ t 1 ( ν ) . Note the kurtosis of t 1 ( ν ) is non-zero, and can be very large or even infinite for small ν . This prediction is validated in Section 4.3. The condition also reveals that not all symmetries in the data (here, elliptical symmetry) induce structure in the trained model weights, if localization is to be seen as a sparsity more structured than oscillatory weights [ cf. God+23]; indeed, translational symmetry (S2) is more relevant for localization than elliptical symmetry.

4 Experimental results

We describe experiments to validate the generalizability of the analytical results from Section 3. We run all experiments on a single CPU machine locally or on a compute cluster. Since all datasets are procedurally generated, training depends on both the model architecture and the complexity of sampling the data, but is between 10 and 60 minutes for any single simulation run.

4.1 Validating Claim 3.2 with positive and negative predictions

In Fig. 5, we validate Claim 3.2 first via the single-neuron model (M2) with 30 initial conditions trained across a range of excess kurtoses for the NLGP ( g ) and Kur ( k ) data models. We use the inverse participation ratio (IPR), defined in Appendix A.3. This measure, also used by Ingrosso and Goldt [IG22], is large when proportionally few weight dimensions 'participate' (have large magnitude), and small when weight dimension magnitudes are more uniform. We see that when g and k assume values that yield a negative excess kurtosis, IPR is close to its maximum of 1 . 0 , suggesting the weights are localized; if the excess kurtosis is positive, IPR is nearly zero, suggesting the weights are not localized. The IPR is extremely consistent across random initializations, suggesting that localization is determined by data statistics and not initialization. The trend in IPR vs. excess kurtosis is very similar between the NLGP ( g ) and Kur ( k ) data models, demonstrating that excess kurtosis is a primary driver of localization and localization is largely independent from other properties of the data distribution.

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Figure 3 further validates Claim 3.2 with specific examples. We maintain constant initial conditions for our model and train on the Ising , NLGP ( g = 0 . 01) , and Kur ( k = 5) data models. The marginals of the Ising model have an excess kurtosis of -2 , the smallest possible value for any distribution. As a result, we see that the amplifier φ for Ising (top left) is super-linear (the dark line exceeds the dashed light line for larger a ), which drives localization via its role in Eq. (5). Integrating Eq. (5) with φ expanded via a third-order Taylor approximation (red line) yields a similar localized receptive field to that from simulation (two right panels), validating this approximation.

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For the remaining distributions (middle and bottom rows) that elicit oscillatory (sinusoidal) weights, Claim 3.2 is validated due to their positive excess kurtosis. The dynamical steady state (far right) assumes a more negative value than in the simulation (to the left), a difference that is the result of deviations of our early-time gradient flow in Eq. (5), but these deviations remain mild enough nevertheless to recover the qualitative structure of the learned receptive field.

4.2 Validating Eq. (5) with localization position prediction

The simulated and integrated receptive fields in Fig. 3 demonstrate that our analytical model is able to meaningfully reproduce localization in receptive fields from neural network training. For the Ising model, we see that the integration even has a peak in the exact same position as the simulation (at index i = 6 ), suggesting precision in our approximation. Indeed, we simulated the condition in Fig. 3 for the Ising model under 28 different initial conditions (weight initializations), and found that in 26 of them (93%), the peaks of the integrated and simulated receptive fields matched exactly. In the two cases where the peaks differed, they did so substantially (see Fig. 8 for an example).

4.3 Validating Proposition 3.3: Elliptical distributions fail to localize

Proposition 3.3 claims that the single-neuron model (M2) trained on elliptical data will yield sinusoidal receptive fields, subject to a condition on the spectra of Σ 0 and Σ 1 . We verify this claim in Fig. 4 with three distinct elliptical distributions. The first, t 40 ( ν = 3) , gives preactivations ⟨ w , X ⟩ that have infinite kurtosis, yet our theory predicts the final receptive field will be sinusoidal. This is confirmed in Fig. 4, where the learned receptive field is indeed a sinusoid with period 1 and intercept at zero.

We also consider data sampled from the surface of an ellipse, which is done by fixing R y ≡ 1 in Definition 1. Here, we observe that the learned receptive field is a near-constant function at -0 . 04 (note that cos(2 π · 0 · x ) ≡ 1 is a sinusoid, allowing nonzero intercepts and constant functions). Finally, we consider an unconventional elliptical distribution where the density of R is given by p R ( r ) = (4 e 2 r +4 ) / ( e 2 r + e 4 ) 2 · ✶ ( r ≥ 2) . This particular density places most of its mass near r = 2 before rapidly falling off, imposing a minimum norm on X and pushing support near the surface of an ellipse. This distribution, too, yields an oscillatory steady state, as shown in Fig. 4 (right). We confirm our visual observations by fitting sinusoids to the final receptive fields and see the relative errors are quite low.

4.4 Extensions to many-neuron model and ICA

All of our analysis thus far has concerned single-neuron models with ReLU activation and without hidden-to-output or bias terms, assumptions which were made to make our analysis tractable. Here, we depart from that regime by considering the SCM and the full two-layer network (Model M1). In Fig. 6 (left) and (center), we train a SCM with 10 hidden units and sigmoid activation on the Kur (10) and Kur (4) datasets, which have excess kurtoses of -0 . 93 and 3 . 28 , respectively. So, based on our single-neuron analysis, we do and do not expect to see localization for these distributions. Indeed, this is precisely what we observe in Fig. 6, where the receptive fields are sharply localized for the former distribution, while they look like low-frequency oscillations for the latter.

magnitude

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w

In Fig. 7, we train many-neuron models with N = 40 input units and K = 10 hidden units, where all weights are learnable. In general, adding flexibility in the second layer leads to more varied structure in the first layer. We train on Kur (4) (top), which has an excess kurtosis of 3 . 28 , and Kur (30) (bottom), which has an excess kurtosis of -1 . 17 . The receptive fields from the former are not localized, as in the singleneuron model; however, they appear more like highfrequency oscillations than low-frequency sinusoids. For Kur (30) , where we expect localization, we see that the first three receptive fields exhibit localization, but less so than for a single neuron. Importantly, not all receptive fields are localized, a result of a variable second-layer weight effectively changing the variance σ 2 in the third-derivative term in Lemma B.1.

We further compare these predictions against ICA, another framework that has been used to model receptive fields in visual cortex. We train on the Kur (3) dataset, which has marginals with excess kurtosis 7 . 66 , fitting 10 components using the FastICA imple- mentation from scikit-learn [ sc ; HO00]. We observe in Fig. 6 (right) that we learn localized receptive fields; this contrasts our neural network models, which require negative excess kurtosis. This stems from ICA's objective to maximize non-Gaussianity, regardless of how specifically it is done. The sign of the excess kurtosis is irrelevant, so long as it is nonzero. This deviation between our analytical model and ICA is an interesting avenue for future study, perhaps by validation with natural images.

5 Conclusions

We derive effective learning dynamics for the minimal example of emergent localization in a neural receptive field given by Ingrosso and Goldt [IG22]. The analytical approach we take relies on the assumption that the conditional preactivation is Gaussian, a refinement of previous work that assumes Gaussianity of the unconditioned preactivation as asserted by the Gaussian equivalence property targeted by Ingrosso and Goldt [IG22]. This approach may prove extensible beyond our specialized setting and may enable further analysis of how statistics of an input task drive emergent structure in neural network learning.

Emergence as an alternative mechanism to top-down constraints like sparsity is in line with recent work that reformulates data-distributional properties as a driver for complex behavior [Cha+22]. Via these analytical effective dynamics, we observe that specific data properties-the covariance structure and the marginals-shape localization in neural receptive fields. Though we cannot capture dynamical interactions between neurons that may shape receptive fields in other settings with the single-neuron analytical model, our empirical validations with many neurons suggest that these interactions do not, in fact, play a significant role in shaping localization [ cf. Har+20].

The data model we consider is a simplified abstraction of the task faced by early sensory systems, and, as a consequence, we do not yet capture certain features of receptive fields that are observed in early sensory systems. In particular, we do not observe orientation nor phase selectivity, features of simple-cell receptive fields in early sensory cortices and in artificial neural networks that can be seen in a subset of receptive fields in Fig. 1 (left and center, respectively). To capture orientation selectivity, it may be fruitful to follow the approach of Karklin and Simoncelli [KS11], who tie orientation selectivity in a population-based efficient-coding framework to the presence of noise. Furthermore, on-center-off-surround-filtering input data, including the idealized data, gives receptive fields with subfields in our simulations, but is difficult to analyze. Lastly, we do not yet look at the distribution of receptive field shapes and do not validate against other models of receptive field learning beyond a brief comparison with ICA [ cf. Sax+11], but these are exciting avenues for future work.

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A Definitions and Notation

A.1 Notation

We use [ n ] to refer to the set { i ∈ N : 1 ≤ i ≤ n } .

A.2 Algebraic sigmoid

For k > 0 , the generalized algebraic sigmoid function is defined as

Following the main text, we drop the subscript when k = 2 .

A.3 Inverse participation ratio (IPR)

The IPR is defined as:

where w i is the magnitude of dimension i of weight w .

B Extensions beyond the scope of the main text

B.1 Analytical properties of the amplifier φ

We present several properties of the amplifying function φ

Lemma B.1. The localization amplifier φ in Eq. (6) satisfies φ ( -a ) = -φ ( a ) , for all a ∈ ( -1 , 1) . Moreover, its derivatives satisfy, where σ 2 and κ denote the variance and kurtosis of X , respectively:

defined in Lemma B.1. 1

To gain some understanding of how the marginal distributions of X impact localization, we use the derivatives in Lemma B.1 to construct a third-order Taylor approximation of φ about 0. The derivatives in Lemma B.1 reveal that every distribution for X 1 with constant variance will look like the same linear function near 0. φ only looks nonlinear once we move sufficiently far away from zero when the third-order term becomes relevant. For the case of σ 2 = 1 (where the variance of X 1 is equal to the value of the larger target y = 1 ) the third order term suggests that φ is super-linear when κ < 3 , i.e. , the excess kurtosis is positive, and sub-linear otherwise.

A super-linear φ will encourage entries where Σ 1 w is large to grow at a faster rate than other entries, which are all subject to the same linear norm constraint through the second term in Eq. (5). The covariance Σ 1 , as a circulant matrix, acts as the convolution operator between some vector and σ 1 1 (the first row in Σ 1 ). Since y = 1 corresponds to the class with a larger length-scale correlation, σ 1 1 will decay relatively slowly and act like a weight local average. Thus, Σ 1 w is the weighted local average for each entry in w . So, entries where w is large within some neighborhood will be encouraged to grow faster than those which are smaller, an effect that compounds as Eq. (5) is integrated. Thus, super-linearity encourages localization.

As we will see in Theorem 3.3 for the setting of elliptical data, if φ is linear, w learns to be sinusoidal, and thus not localized. In the case of sub-linearity, we expect suppression of larger values, rather than promotion, as in the super-linear setting. Thus, to a first approximation, the sign of the excess kurtosis, κ -3 (for σ 2 = 1 ), indicates whether w localizes.

However, simply studying φ ′′′ (0) is not sufficient to fully characterize how the marginals impact localization. A function can be sub-linear for small a and super-linear for larger a , making it unclear whether it will yield localization. For marginal distributions where κ ≈ 3 that do not exhibit strict super- or sub-linearity, this condition is no longer precise enough to determine whether we see localization.

B.2 PDE limit of Eq. (5)

By taking N to be large and treating w as a continuous function with respect to position, i.e. , w ≡ w ( x, t ) , one can treat Eq. (5) as a partial differential equation (PDE). Finding its steady state then amounts to solving

where w : [0 , 1] → R is periodic and σ y is the convolution corresponding to the limiting case of the matrix Σ y . This equation does not appear to have an explicit solution for non-identity Σ 1 , and thus, it may not be possible to solve the steady states of Eq. (5) exactly in this PDE limit or for finite N .

B.3 Assumptions A1 and A2 vs. Gaussian equivalence

Assumptions A1 and A2 are equivalent to approximating ⟨ w , X ⟩ | X i as Gaussian early in training. Similar ideas have been used to derive gradient flow dynamics for neural networks, including in developing the Gaussian equivalence property of [Gol+20; Ger+20; Gol+22]. However, these works model the unconditional preactivation ⟨ w , X ⟩ as a Gaussian, rather than first conditioning on X i . How this arises is that these previous works assert the Gaussian approximation prior to differentiating the loss function for the gradient flow dynamics. However, an approximation at that stage neglects the presence of a multiplicative factor X that appears as a result of the chain rule applied to ⟨ w , X ⟩ . Abstractly, this approach assumes that L exact → L Gauss implies ∇ w L exact → ∇ w L Gauss , but, in general, this does not follow, and here in particular this assumption does not capture the interplay of learning and higher-order input statistics. This contributes to the failure of Gaussian equivalence in [IG22]. In contrast, we can account for the additional X term in the derivation of Lemma 3.1 by assuming that ⟨ w , X ⟩ | X i rather than ⟨ w , X ⟩ is Gaussian. This conditioning approach, along with the translation invariance of the data (Property S2), also motivates considering the marginal distributions X i as the object of study to obtain gradient flows for neural networks trained on non-Gaussian inputs.

C Proofs of theoretical results

C.1 Gradient flow for mean-squared error (MSE) loss

The loss is given by:

The assumption of sign symmetry (S3) gives that ⟨ w , X ⟩ is also sign-symmetric. First, this implies that /a80 ( ⟨ w , X ⟩ > 0) = 1 2 , so: ( II ) = 1 2 E X ,Y |⟨ w , X ⟩≥ 0 [ ReLU 2 ( ⟨ w , X ⟩ ) ] = 1 2 E X ,Y |⟨ w , X ⟩≥ 0 [ ( ⟨ w , X ⟩ ) 2 ] .

Second, sign-symmetry of ⟨ w , X ⟩ implies that we can drop the conditioning on ⟨ w , X ⟩ ≥ 0 , since ⟨ w , X ⟩ d = -⟨ w , X ⟩ . Thus,

where K is the number of values (classes) of discrete y . Finally, we differentiate L with respect to w :

The gradient flow [EC24] is given by d w d t = -τ ∇ w L , where τ is the learning rate. Thus,

C.2 Proof of lemma 3.1

Setting K = 2 in equation (10), we have

We wish to express the first term explicitly. Note that the first term is a vector in R N . We consider each of its entries separately by using the law of total expectation to write the i -th entry as:

By Assumption A3, { w i X i | 1 ≤ i ≤ N } satisfies Lindeberg's condition as N → ∞ . This is also known as a uniform integrability requirement. Before formally stating it, let us introduce two variables: S N ≜ ∑ N j =1 w j ( X j -µ j | x i ) , the partial sums, and their variance, σ 2 N ≜ E [ S 2 N ] , where µ j | x i ≜ E [ X j | X i = x i ] is the conditional mean of the j -entry given that i -th entry has value x i . Then, Lindeberg's condition is formally stated as

This condition effectively states that no term in the partial sum w i X i will dominate. Under this condition, along with weak dependence (Property S1), we conclude from [Bra07, Theorems 1.19, 10.2] that S N /σ N d →N (0 , 1) . Note that

where w , X are N -dimensional vectors, µ | x i = E [ X | X i = x i ] is the vector of conditional means, and Σ 1 | x i ≜ Cov [ X | X i = x i , Y = 1] = Σ 1 -σ 1 i σ 1 ⊤ i . Since σ N and µ | x i are constant, we can write

where the second step, in which we acquire o N (1) , follows from the definition of convergence in distribution. Under Assumptions A1 and A2, we may express this as

Therefore,

Defining φ i ( a ) ≜ E X i | Y =1 [ X i erf( X i alg -1 ( a ) / √ 2)] we can write

Note that E X i [ | X i | ] ≤ √ E X i [ X 2 i ] = σ by Cauchy-Schwarz. So, E X i [ | X i | ] o N (1) = o N (1) . Moreover, by translation-invariance, all X i have the same marginal, so φ i ≡ φ 1 ≜ φ . Thus,

This form holds for all entries i . Concatenating them, we obtain

which gives the desired result.

C.3 Proof of lemma B.1

Property 1 follows from the fact that alg and erf are odd functions. This implies Property 4 since an odd function must have zero for even Taylor coefficients.

Properties 2 and 3 follow from differentiating Eq. (6). The first derivative is given by

Setting a = 0 , we get

The third derivative is given by

Again setting a = 0 gives

C.4 Proof of Proposition 3.3

The pdf of X ∼ E N ( µ, Σ , ϕ ) is

for some function g : R ≥ 0 → R ≥ 0 [Fra04]. A key property of elliptical distributions is that if X ∼ E N ( µ, Σ , ϕ ) , then its affine transformation is also elliptical: ⟨ w , X ⟩ ∼ E 1 ( ⟨ w , µ ⟩ , w ⊤ Σ w , ϕ ) for any w ∈ R N . Thus,

for some other function ˜ g : R ≥ 0 → R ≥ 0 .

From our assumption of sign-symmetry, we have µ = 0 . For brevity, we define σ 2 ≜ w ⊤ Σ w and S ≜ ⟨ w , X ⟩ . We begin by computing (I) in Eq. (9). Recall that we have y = 0 , 1 ( i.e. , K = 2 ). So,

At this point, we apply a u -substitution with u = s 2 /σ 2 , and thus d u = 2 s d s/σ 2 ⇐⇒ σ d u/ 2 = s d s/σ . This yields

Recall that we assume the MSE loss L is finite. The first term in Eq. (9) is clearly finite for y = 0 , 1 . The term ( II ) is also easily seen to be finite, since it evaluates to w ⊤ (Σ 0 +Σ 1 ) w / 4 . Thus, L being finite implies ( I ) in Eq. (9) is finite, which implies C < ∞ . We computed (II) from Eq. (9) above as

for K = 2 . Plugging in (I) and (II) and differentiating with respect to w , we get

The steady states of Eq. (11) thus satisfy

Recall that translation-invariance implies that Σ 0 and Σ 1 are circulant, and since they are covariance matrices, they are symmetric. Then, they diagonalize in the basis given by the real and imaginary parts of the first n/ 2 Fourier components in the discrete Fourier transform, which we denote by the n × n real matrix F . Note that F is orthogonal. Define v = F ⊤ w and Λ y = F ⊤ Σ y F for y = 0 , 1 . Thus, the steady states satisfy

This holds iff for all i ∈ [ n ] ,

̸

Thus, if v i = 0 , we must have that

That is, the ratio of the i -th eigenvalues of Σ 0 and Σ 1 must be constant for all i s.t. v i = 0 . The eigenvalues of these matrices always come in pairs because of how we defined F using both the real and imaginary parts of the discrete Fourier transform. In general, we observe that each pair assumes a unique value. So, since C is finite, the condition above can hold for at most two distinct values of i . Therefore, v i = 0 for all but at most two i ∈ [ n ] , implying that the steady state w is of the form a cos(2 πkx ) + b sin(2 πkx ) , i.e. , it is oscillatory. As such, it is not localized.

D Additional experiments

D.1 Visualizing breakdown of Assumption A3

Fig. 8 demonstrates that our analytical model holds for long enough during training to capture the emergence of localization in the single ReLU neuron (M2). In the first three columns, we visualize the IPR of the weights from our empirical and analytical models, as well as the ℓ 2 difference between these two weights. In the first four rows, we visualize these metrics for four random initializations of the model, training each on NLGP ( g = 100) with ξ 0 = 0 . 3 and ξ 1 = 0 . 7 , where we expect from Theorem 3.2 to see localization. We see the error rapidly increase shortly after IPR increases, indicating the formation of localized receptive fields. The last three columns confirm this, as they show snapshots from before , during , and after the divergence between the empirical and analytical weights. We observe the weights are nearly identical before , differ only slightly at the most localized point during , and are both localized after , but possibly with different magnitudes and positions. The difference that emerges during is due to a breakdown of Assumption A3 used to create our analytical model, which is violated when the norm of w is dominated by just a few entries, i.e. , it is localized. While [IG22] also observe a breakdown in their analytical model as localization emerges, ours, crucially, holds for long enough to characterize the emergence of localization.

We discuss the individual subplots in more detail. In all but the third row of Fig. 8, the analytical predictions are near-exact; in the third row, we predict localization, but at the wrong position. Focusing again on the first row, we see that at t = 20 , the weights have not yet become localized (from IPR, left, first, and visually) and analytical and empirical weights match nearlt exactly, as confirmed by the small distance in left, center above. At t = 30 , a localized peak around i = 21 begins to emerge, violating Assumption A3 and weakening analytical precision. The analytical model then underestimates the degree to which the main peak at i = 21 dominates, while it overestimates the size of competing peaks at i = 30 , 37 , and 90 . Despite this, at t = 50 , we see that predictions from the analytical model retain a match to the empirical model.

In the last row of Fig. 8, we use the same initialization and setting as in the first row, except that we train on NLGP ( g = 0 . 01) data instead. From Theorem 3.2, we do not expect to see localization. The evolution of IPR confirms this, as it stays low in magnitude. We also see that, because localization never emerges, Assumption A3 is never violated, and so our analytical model accounts for the empirical model nearly perfectly.

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NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: Yes, in the abstract and introduction we claim to derive an analytical model for the effective dynamics of localization (Section 3) in the setting described in Section 2 and validate this model with experiments (Section 4).

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: Yes, in the conclusion we note that our exact analysis is retricted to the singleneuron architecture M2 in the data setting described in Section 2, and that validations beyond that scope (like in Section 4) remain empirical even if they show promise.

3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: Yes, our assumptions are rigourously stated in Section 3 and the proofs are provided in Appendix C.

4. Experimental Result Reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: Our setup (Section 4) and experiments (Section 4) provides full details on the data, hyperparameters, and training procedures used to obtain the figures. We also provide code (referenced on the first page).

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: Yes, we provide a link to the code repository in the first page of the paper, and the repository contains all the code and data needed to reproduce the experiments.

6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: Yes, we provide primary details in Section 4 and all the details in the code.

7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: Yes, for our prediction experiment in Section 4.2 we report a measure of variation over multiple runs.

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Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

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Answer: [Yes]

Justification: We wrote our own code for all experiments to procedurally generate data and train and analyze models. Visualizations from prior work in Fig. 1 are properly credited.

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Answer: [Yes]

Justification: We document our code (the only asset we release) in the accompanying repository.

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Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates when localized receptive fields arise in supervised neural networks. Extending a recent work of Ingrosso and Goldt, the authors propose that simple single-neuron models learn localized receptive fields when trained on data with sufficiently negative excess kurtosis, while if the excess kurtosis is sufficiently positive they learn delocalized receptive fields.\n\nSTRENGTHS:\nThe topic of this paper is of broad interest in both machine learning and neuroscience, and on the whole I think this manuscript makes a worthy contribution on top of the work of Ingrosso and Goldt. There are some weaknesses which dampen my enthusiasm (see below), but on the whole I favor acceptance.\n\nWEAKNESSES:\nI have two primary concerns about the results presented:\n\n- First, Lemma 3.1's treatment of time is not sufficiently precise. Can you provide a more precise answer than simply \"early in training\" or \"before A.3 is violated\"? Indeed, the logic in the paragraph beginning on Line 174 is not clear. What you show is that, in some cases (see concern below), eq. (5) generates localized RFs in a similar location to those observed in actual training. This does necessarily mean that \"the gradient flow in Eq. (5) holds sufficiently long to detect the emergence of localization in the weights,\" as you write in Lines 177-178. Moreover, you have not in fact defined what you mean by \"detect the emergence of localization;\" this must be reified. Can you see a change in participation ratio, even if only numerically, before the approximation breaks down? \n\n- Second, the experiments are rather limited, and rely largely on exemplars rather than systematic statistical investigation. This is important given the gap in Lemma 3.1: the authors rely on experiments to justify their claim that this approximation provides meaningful information about when localization will occur, but all they actually show is that there is resemblance in a few cases. The paper would be much stronger if the authors could also show that their claims hold statistically over many realizations of the data generation process and training procedure.\n\nQUESTIONS:\n- A neuroscientific quibble: in the first sentence of the Introduction (Line 19), one need not restrict attention to the \"mammalian nervous system.\" There are numerous examples of localized receptive fields in non-mammalian species; see for instance the beautiful works of Knudsen & Konishi on audition in owls. \n\n- The authors might consider citing Sengupta et al., \"Manifold-tiling Localized Receptive Fields are Optimal in Similarity-preserving Neural Networks\" (NeurIPS 2018) in their discussion of unsupervised learning algorithms that give localized receptive fields. \n\n- Another potentially relevant reference is Shinn, \"Phantom oscillations in principal component analysis\" (PNAS 2023). \n\n- The reference in Footnote 3 is wrong; it should be to Appendix C.2 not C.3. \n\n- In Line 187, there is a small typo: \"termdepends\" -> \"term depends\"\n\n- Using a perceptually uniform non-grayscale colormap to represent time might improve the legibility of the plots relative to the grayscale used in the submitted manuscript. \n\n- The final paragraph of the conclusion is not tied to anything that came before; if you care about orientation selectivity you should mention & measure it in earleir portions of the paper. Otherwise, this should be omitted.\n\nLIMITATIONS:\nThe authors do a largely adequate job of discussing the limitations of their work, up to the technical weaknesses noted above.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nSETTING: Olshausen & Field famously showed that requiring natural image patches to be constructed from a small number of independent elements from an overcomplete dictionary populates that dictionary with spatially localized feature detectors.  But localization also appears in DNNs trained on discriminative-learning tasks.  It is also ubiquitous in the cortex.  This raises the question of what fundamentally drives the formation of localized receptive fields.\n\nAPPROACH: The MS attempts to explain the emergence of localized RFs by reference to the statistics of the stimuli (inputs).  In particular, the authors examine the time evolution of network weights trained with gradient descent to discriminate \"natural-image-like\" stimuli--analytically for a single (ReLU) unit of neural network, and in simulation for a weighted sum of multiple such units (a two-layer NN with one output unit).\n\nMore precisely, the authors consider stimuli/inputs with circulant covariance matrices, a 1D idealization of natural images (the covariance matrices of natural images are block-circulant with circulant blocks).  Under some additional simplifying assumptions (see below), they derive an ODE for the time evolution of the weights when the network is trained to discriminate stimuli according to whether their spatial correlations are \"long\" or \"short.\"  They show that this ODE depends only on the marginal distribution of a single \"pixel\" rather than the joint (and is valid until localization begins).  Investigation of this ODE (Appendix C.1) reveals that localization will occur (w large for some entries, small for the rest) when the inputs have (sufficiently) negative excess kurtosis.  The authors also demonstrate this in simulation with data distributions with varying levels of excess kurtosis, both by examining the learned receptive fields and by numerically integrating the ODE.  Finally, they extend their simulations to two-layer NNs.\n\nSTRENGTHS:\nThe MS persuasively presents a new route to localized receptive fields.  The proofs look correct (but I did not verify every line) and the simulations confirm that they are not undermined by the various approximations employer along the way.  The problem is an important one and of long-standing interest to the field (computational neuroscience).\n\nWEAKNESSES:\nThis reviewer struggles to see how these results can be generalized beyond this very simplified setting of, essentially, a \"network\" Y = ReLU(<w, X>) (or its cousin, the SCM).  Introducing just one more learnable layer breaks (or at least vitiates) the connection between stimulus kurtosis and localized receptive fields (as the authors show in Fig. 6).\n\nThis makes it hard to find this explanation of RF localization more compelling than the classical one in terms of efficient codes.\n\nQUESTIONS:\n--One of the chief motivations for the study is that even DNNs trained on supervised-learning tasks acquire localized RFs.  In support of this claim they cite the AlexNet paper and some papers on visualizations of convnets.  Obviously the convolutions in these networks enforce spatial localization.  Do the authors have in mind here localized (bandpass) *frequency* RFs?\n\n--Is *negative* excess kurtosis consistent with the statistics of natural scenes?\n\n--Can the authors provide more intuition about assumptions A1 and A2, and how restrictive they are?\n\n--In the proof, l. 634, shouldn't it be P(<w,X> > 0) = 1/2?  (There is no > 0 in the expression in the MS.  Am I misunderstanding this?)\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors analytically derive the learning dynamics of (extremely) simple neural network models and characterize conditions under which units learn localized receptive fields. This builds on celebrated work in computational neuroscience on sparse coding and independent components analysis, offering a new perspective for biological findings.\n\nSTRENGTHS:\nThe paper is beautifully written (but see comments about figures below). It is well-motivated and accessible to a broad audience. The target audience for this kind of work may be somewhat niche, but it fits within the scope of the NeurIPS conference. The authors summarize prior work concisely and clearly. The assumptions of the analysis are stated precisely. The proofs in the appendix are well written, and the main claims of the theory are tested experimentally (though see below for some areas I am a bit unclear on).\n\nWEAKNESSES:\nTo enable analytic tractability, the paper relies on very strong and simplifying assumptions. Figures 4, 5, 6, and the right hand side of Figure 3 are hard to see. It could be helpful to add color and show fewer lines. Also, the description of these figures is the one part of the text I had trouble following. Quantifying the outcomes somehow would be helpful for me to understand exactly what I'm supposed to be looking at in these examples. For instance, in Figure 4A, the claim is that the model learns an oscillatory filter that resembles a sinusoid. But the left panel in Fig 4A, though oscillatory, doesn't look sinusoidal?\n\nUltimately I believe these weaknesses can be addressed during the rebuttal phase and the strengths outweigh the weaknesses.\n\nQUESTIONS:\n- In addition to showing examples in Figures 4, 5, and 6, can you somehow quantify localization?\n\nLIMITATIONS:\nLimiting assumptions are clearly stated and discussed.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-11"}