Update after rebuttal ---------------------------- Thanks the the authors for their response. I have read their comments, but have decided to maintain my original score, as my concerns about the input noise and correlations largely remain. Original review -------------------- I enjoyed reading this paper. The paper leverages the fact that for a block diagonal input covariance matrix (which occurs if the inputs to the two regions are independent), then the optimal solution relies on simply taking the top $m$ eigenvalues from the jointly sorted list--this means that the output space can have proportionally different representations of the input, compared to the original ratio (called $r$ in the paper). This is a simple, yet surprising, fact that is clearly presented. The mathematical details are also easy to follow. I am unaware of other work in the efficient coding literature that addresses this problem in the same way, as such, I believe the work is original. The figures are clean and elegant and also greatly aid the presentation. I would have liked to see the authors address a couple of additional scenarios in their work: - What if there are not just two different receptor densities, but three, four, or many? In that case, what does the resource allocation look like. This is mentioned in the discussion, but it seems like a simple enough extension that may be worth including in this paper. - What about input noise? Decorrelation is optimal only when there is no input noise. The presence of (independent) input noise means that the system may want to average over the redundancies in the inputs to improve the signal to noise ratio. Can this analysis be extended to this scenario (c.f. Doi et al, ref. [6] in the paper). - I think more should be said about the block diagonal assumption. It seems unlikely that nearby receptors (fovea vs periphery, or fingertip vs hand) will experience totally independent inputs (there are strong long range correlations in natural stimuli). I appreciate that the assumption allows for an analytic solution, but perhaps more can be said about how the analytic solution breaks down as this assumption is violated. - There is a difference between receptor densities that tile *different* parts of the input space (e.g. cones in the fovea vs periphery, or mechanoreceptors in the fingertip vs the rest of the hand), vs receptors that tile the *same* part of the input space (e.g. overlapping cell types in the retina, or multiple mechanoreceptor types in the skin). The paper seems to address the former scenario, where correlations between the receptors would be reduced, but what about the latter? Perhaps the authors can comment in the discussion on whether this analysis is appropriate for overlapping receptor populations that each tile the same input space.

Major: What is the explicit objective function: reconstruction, information preservation? In what noise regime are we? input noise, output noise, large, infinitesimal? My understanding is that the objective function is information preservation in the infinitesimal noise regime for both input and output. The objective function should be stated explicitly, and the equations of the article should ideally be derived from this initial objective function. Minor "In vision, the density of cones in the retina differs by several orders of magnitude between the fovea and the periphery" => closer to one order of magnitude: see https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4985666/ eq7: define a, gamma, x. where does the number of receptors appear? eq8: how did you apply the Fourier transform? steps missing. eq10: what is the unit of a? what is the number of receptors? Do you assume that both regions have the same size (low density and high density regions)? Possible discussion point: why are receptors arranged in an inhomogenous way in the first place? clarity: OK quality: good. originality: haven’t seen anything like this. significance: medium/high. implications for AI?

The paper considers an efficient coding setup in which the input receptors are non-uniformly distributed. The optimal neuronal allocation is derived based on varying sensory bottleneck and input density. To my knowledge, this particular setup has not been carefully studied in the previous work. Overall the study is well executed and the logic is clear. The method used is also technically sound. The paper is generally well written, and relatively easy to follow. I have two main concerns: First, the current model setup ignores many critical ingredients, such as noise in the output, noise in the input, output nonlinearity, and the metabolic cost. It is known that these factors can fundamentally change the optimal coding configuration. The model presented currently is a linear model without output (spiking) noise and metabolic cost. It is difficult to judge how these additional ingredient might change the solution derived in the present work. This is an important issue because it affects the interpretation and the potential contribution of this work, in terms of how well the theory could explain the neurophysiology, as well as whether the regime considered in the paper is a relevant regime for understanding the physiology. Second, while the theoretical derivations are neat and the analytical results are interesting, the connections to experiments are rather weak, at least as it stands now. The application to the natural images statistics with the 1D model is helpful, but the relevance to the physiology is largely unclear. This makes the significance of the work uncertain. More specific comments: Concerning modeling the non-uniform input- Does input X contain any input noise? If so, how does the non-uniform input density affect the characteristics of the input noise? If not, how would adding input noise change the results? It is not obvious how receptor density is modeled in Eq (6). It would useful to clarify. Line 26- it would useful to cite previous work to support the claim here. Line 188,189- it would be useful to cite the relevant experimental literature here. The title doesn’t seem to fully capture the gist of the paper; I’d suggest something like ”Optimal resource allocation in sensory bottlenecks with non-uniform input sampling”. Of course, this is entirely up to the authors.. ** edit after the rebuttal I found the authors' response and the discussion with other reviewers to be helpful. Although I still like to see more ingredients such as input/output noise, metabolic constraints, nonlinearity, to be added to the theory to make it more relevant for neurophysiology, at the same time, I think the current version is already interesting enough. Although I am not increasing my score here, I'd like to vote for acceptance of this paper.