Nonlinear dynamics of localization in neural receptive fields

Summary

This 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.

Strengths

The 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.

Weaknesses

I have two primary concerns about the results presented: - 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? - 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.

Questions

- 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. - 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. - Another potentially relevant reference is Shinn, "Phantom oscillations in principal component analysis" (PNAS 2023). - The reference in Footnote 3 is wrong; it should be to Appendix C.2 not C.3. - In Line 187, there is a small typo: "termdepends" -> "term depends" - 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. - 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.

Rating

7

Confidence

4

Soundness

2

Presentation

3

Contribution

3

Limitations

The authors do a largely adequate job of discussing the limitations of their work, up to the technical weaknesses noted above.

Summary

SETTING: 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. APPROACH: 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). More 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.

Strengths

The 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).

Weaknesses

This 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). This makes it hard to find this explanation of RF localization more compelling than the classical one in terms of efficient codes.

Questions

--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? --Is *negative* excess kurtosis consistent with the statistics of natural scenes? --Can the authors provide more intuition about assumptions A1 and A2, and how restrictive they are? --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?)

Rating

8

Confidence

4

Soundness

4

Presentation

3

Contribution

3

Limitations

N/A

Summary

The 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.

Strengths

The 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).

Weaknesses

To 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? Ultimately I believe these weaknesses can be addressed during the rebuttal phase and the strengths outweigh the weaknesses.

Questions

- In addition to showing examples in Figures 4, 5, and 6, can you somehow quantify localization?

Rating

7

Confidence

3

Soundness

4

Presentation

3

Contribution

3

Limitations

Limiting assumptions are clearly stated and discussed.

Thanks for the reply!

Thank you for your thorough reply to my comment. I think you've adequately addressed my comments and those of the other referees, so I will raise my score as I think this paper should be accepted.

Reviewer Response

Thanks for the clarifications. The sinusoids are still a bit hard to see, but the new firgure is an improvement. Perhaps an "approximate" sinusoid. Overall I think this is minor; I retain my positive score.