Nonlinear dynamics of localization in neural receptive fields

We thank the reviewer for their extremely helpful feedback, in particular for making us aware of relevant work and clarifying minor misconceptions. We also appreciate your insistence to more comprehensively quantify and replicate our analyses.

Lemma 3.1's treatment of time is not sufficiently precise.

We thank the reviewer for this criticism. Indeed, lemma 3.1 does not provide a precise criterion for when the approximation breaks down. This is because the only assumption that is violated as a result of localization, A3, does not break down at any predictable $t$. To make this claim clearer, we have included Fig. 2 & 3 in the Global Response. We see across examples that the analytical model closely tracks the empirical one until a strongly localized bump emerges in the weights; when no localization emerges, the analytical model maintains its precision through training. Please see the figure and its caption for more detailed information. We would happily incorporate suggestions from the reviewer on how best to represent or quantify this notion of time.

the experiments are rather limited, and rely largely on exemplars rather than systematic statistical investigation

We thank the reviewer especially for stating this limitation. To address this, as well as other reviewers' concerns about the precision of our model and analysis, we have included Fig. 1 in the Global Response. This figure shows the IPR against the excess kurtosis for various values of $g$ and $k$, respectively, for $`NLGP`$ and $`Kur`$.

one need not restrict attention to the "mammalian nervous system"

Thank you for the reference; Knudsen & Konishi (1978) is indeed relevant. We will broaden our terminology from "the mammalian nervous system" to "animal nervous systems".

Knudsen & Konishi (1978). Center-surround organization of auditory receptive fields in the owl. DOI:10.1126/science.

consider citing Sengupta et al.

Thank you for the reference; we will add Sengupta et al. (2018) to the list of unsupervised learning algorithms producing localized receptive fields. As far as we understand, this algorithm does not optimize an explicit or implicit sparsity criterion as in sparse coding, so it is also an example of an alternative model for the emergence of localization. $Har+20$ cited in the submission is another such example; we will incorporate both of these references in the submission via discussion of their modelling assumptions wrt those we employ.

The final paragraph of the conclusion $on orientation selectivity$ is not tied to anything that came before ...

We are asked by NeurIPS to address the limitations of our work, and since we display oriented receptive fields in Figure 1 (left, center) we would like to call attention to the fact that we cannot yet capture this feature in our analytical model, and neither do the simulated receptive fields appear oriented. To improve on this point, we will introduce the terminology of oriented receptive fields when referring to Figure 1 (left, center) in the introduction so that this terminology does not spring out of nowhere in the conclusion, as the reviewer has pointed out. Our analytical and simulated receptive fields are not oriented to any degree, so we do not think it is necessary to explicitly measure this property.

typo, corrections

us $e$ a perceptually uniform non-grayscale colormap

We particularly appreciate the reviewer's thorough examination of our work and their identification of typographical errors. A compiled list of the typographical corrections we plan to implement is provided in the global response. We also intend to reformat our plots to use a non-grayscale colormap to improve legibility, which another reviewer encouraged as well. Please see Fig. 4 in the attached PDF for an example.

We thank the reviewer for their excellent feedback. We especially appreciated your precise and astute questions with regard to the validity and generalizability of our model, and also for reading closely enough to identify a typo in our proof.

how these results can be generalized beyond this very simplified setting ... one more learnable layer breaks (or at least vitiates) the connection between stimulus kurtosis and localized receptive fields

The reviewer correctly points out a key limitation of this work: extensions to more complicated models quickly lead to difficulties in extending our analytical approach and its interpretation. Indeed, we observe that experiments with two-layer networks are noisier (as the reviewer pointed out in referencing Figure 6), as did [IG22].

However, we consider it worth mentioning that some aspects of our analytical approach seem to extend to the two-learnable-layer regime, though we reserve this as an avenue for future work. As an example, using the approach we have developed, consider the steady-state equation for a single-neuron model with two learnable layers, $\mathbf{w}_1$ and $w_2$, after plugging in for $w_2$: $\varphi\left( \frac{\Sigma_1 \mathbf{w}_1}{\sqrt{\langle \Sigma_1 \mathbf{w}_1, \mathbf{w}_1 \rangle}} \right) = \frac{1}{\sqrt{2 \pi}} \frac{(\Sigma_0 + \Sigma_1) \mathbf{w}_1}{\sqrt{\langle (\Sigma_0 + \Sigma_1) \mathbf{w}_1, \mathbf{w}_1 \rangle}}$

This steady state is very similar to the one implied by Eq. 5 in Lemma 3.1, the only difference being an additional scaling term in the denominator of the right side. We have found that manipulating this term does not substantially change whether a given $\varphi$ yields localized receptive fields, but this observation is still preliminary. However, dynamics play a very important role in the precise structure obtained at convergence, something we are exploring for potential future work. As the reviewer points out, adding another learnable layer obfuscates the relation between stimulus kurtosis and localization because the second-layer weight can be pulled into ReLU term and viewed as a rescaling of the stimulus. A thorough analysis will likely not be as simple as just studying $\varphi$ and will likely require a careful analysis of initializations, but we expect that general strategies and ideas from this work will carry over into future ones.

hard to find this explanation of RF localization more compelling than the classical one in terms of efficient codes

We would like to emphasize that we don't wish to claim that our approach is much more compelling than the sparse coding work of [OF96]. Instead, we hope to present it as an alternative bottom-up, learning model that should be investigated further because of its ability to reproduce key qualitative phenomena of visual receptive fields with an alternative set of assumptions that some may view as more minimal (due to lack of sparsity regularization).

the convolutions in these networks $such as AlexNet$ enforce spatial localization

The kernel size of the first-layer convolutional kernels is indeed usually taken to be much smaller than than the size of the input image, thus enforcing a degree of spatial localization. For AlexNet, for example, receives input image of size $224 \times 224$ with an $11 \times 11$ kernel in the first layer. Here, we mean to refer to the further localization that emerges within these convolutional kernels within that kernel bandwidth; i.e., the oriented receptive fields visible in Figure 1 have width in the edge direction much smaller than 11.

Is negative excess kurtosis consistent with the statistics of natural scenes?

While we focus on analyzing the model of emergent localized receptive fields of [IG22], this is a great question about broader relevance. We can provide some context in the case of simple cells in primary visual cortex (V1), of which sparse coding is classically taken as a model [OF97] Retinal ganglion cells prior to V1 are understood to perform edge detection. Edge detection naturally induces concentrated marginals, corresponding to positive and negative edges, with a large amount mass at zero, corresponding to no edge. Such distributions typically have negative excess kurtosis (unless a very large amount of mass is at zero, corresponding to a nearly uniform input).

Can the authors provide more intuition about assumptions A1 and A2, and how restrictive they are?

We thank the reviewer for this question. We attempted to address in lines 160-3 in the submission, which we expand on here.

A1 and A2 are motivated by the limiting cases of the $`NLGP`(g)$ data model from [IG22]: $g \to 0$ (no localization) and $g \to \infty$ (localization). A1 is implied by the weaker assumption $\mathbb{E}[X_j \mid X_i = x_i, Y = y] \propto x_i$ after applying S3. A2 captures that when conditioning on $X_i$, nearby entries in $\mathbf{X}$ are almost deterministic, while distant ones are unaffected. The $i$-th row and column of $\operatorname{Cov}[X_j \mid X_i = x_i, Y = y]$ are zero. Local dependence implies entries $(k,j)$ of the conditional covariance for $k,j$ near $i$ should be small, while distant ones remain unchanged. Weak dependence (S1) implies $\sigma_i^y$ is largest near $i$ and zero elsewhere. Subtracting $\sigma_i^y {\sigma_i^y}^\top$ from $\Sigma_y$ expresses these intuitions, supported by $`NLGP`$ limiting cases. As such we consider these assumptions to be relatively loose. Fig. 1 of the Global Response, which shows a sharp uptick in IPR as excess kurtosis becomes negative, supports this.

... l. 634 shouldn't it be P(<w,X> > 0) = 1/2?

Yes; thank you for the correction.

We thank the reviewer for their well-formulated concerns and suggestions, especially with regard to improving the interpretability and readability of our work.

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.

We thank the reviewer for the suggestion to improve legibility of the receptive field evolution. In the Global Response, we have plotted receptive fields with a non-grayscale (blue-red) colormap to improve legibility, which another reviewer suggested as well.

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. in addition to ... Figures 4, 5, and 6, can you somehow quantify localization?

We thank the reviewer for the suggestion to quantify the localization phenomenon. In Fig. 1--3 of the Global Response, we quantify localization with the inverse participation ratio (IPR), defined as $\operatorname{IPR}(\mathbf{w})=\left(\sum_{i=1}^D w_i^4\right)/\left(\sum_{i=1}^D w_i^2\right)^2$, where $w_i$ is the magnitude of dimension $i$ of weight $\mathbf{w}$. This measure, also used by [IG22], is large when proportionally few weight dimensions "participate" (have large magnitude), and small when weight dimension magnitudes are more uniform.

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?

In Fig. 4 of the Global Response, we quantify this claim by fitting a sinusoid to the resulting receptive field, and find that we can do so very well. We also rescale the receptive field to make the sinusoidal structure of the steady state more apparent. Please see the figure caption for more details.