{"id":"03bde80c-67f0-4fb4-b569-6cf7eeb32625","arxiv_id":"2607.07470","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":3,"one_line_summary":"For stationary image datasets and standard augmentations, the optimal contrastive representation is partial whitening of DFT power, implemented by a CNN with sinusoidal first-layer filters and a waterfilling weight allocation.","lead":"This paper proves that the optimal representation for contrastive learning on stationary image datasets is partial whitening of Fourier power spectra, computable by a simple CNN with sinusoidal filters. It explains why contrastive learning works even with synthetic noise images: the power spectrum, not semantic content, determines the learned features.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The Gaussianity of y(x) is the linchpin connecting GUPA theory to InfoNCE practice, validated only empirically. A direct comparison of the two losses for the predicted optimal representation would settle whether the bridge holds.","rationale":"The reader's verdict of CONDITIONAL with MODERATE confidence is appropriate. The theorems about GUPA are mathematically sound — the proofs rely on well-established results (Theorem 2.1 from Peligrad & Wu 2010) and the algebra in Appendices D and G is correct. The waterfilling optimization (Theorem 1.3, Algorithm 1) follows standard convex optimization arguments. The experiments provide reasonable empirical support: sinusoidal filters are learned across datasets and augmentations (Figures 2, 7, 8), partial whitening is observed (Figure 6), and InfoNCE and GUPA give similar results (Figures 11–13). The CONDITIONAL verdict correctly reflects that the Gaussianity assumption (Theorem 1.1) is unproven and is the main barrier to claiming the results apply to standard InfoNCE-based contrastive learning. The paper is commendably honest about its limitations (Section 5), explicitly listing the GUPA vs. InfoNCE gap, the simplified architecture, and the crop translation approximation. The novelty is genuine — connecting stationarity, DFT independence, and waterfilling to contrastive learning is a new and insightful theoretical contribution. No code release is a minor concern but does not affect the theoretical claims. I agree with the reader that the Gaussianity assumption is the weakest link, and I propose a concrete test that would directly measure whether the GUPA-InfoNCE gap is significant for the predicted optimal representation. If that test shows the losses are close, the CONDITIONAL could be upgraded to ACCEPT; if they diverge, the practical relevance of the theorems would be significantly weakened. The verdict should remain UNCHANGED as CONDITIONAL.","tokens_in":23690,"tokens_out":6467,"duration_ms":379239,"concrete_test":"For the analytically predicted optimal representation y*(x) from Theorem 3.1 (squared DFT coefficients with inverse-variance scaling) computed on CIFAR10, directly evaluate both the InfoNCE loss (Eq. 1, with large batch) and the GUPA loss (Eq. 2) on the same representation. If the two losses differ by more than a few percent relative to their scale, the Gaussianity assumption is violated for this representation and the GUPA→InfoNCE bridge is unreliable. Additionally, apply a multivariate normality test (e.g., Henze-Zirkler) to y*(x) over the dataset to quantify the degree of non-Gaussianity. This directly tests Theorem 1.1's condition for the specific representation the theory predicts is optimal.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central theorems (3.1, 3.2) are mathematically correct for the GUPA loss — the proofs via diagonal Σ (DFT independence, Theorem 2.1) and diagonal B (per-frequency augmentation action, Appendix G) are sound. The load-bearing concern is not about these theorems themselves but about Theorem 1.1, which equates GUPA to InfoNCE only when y(x) is Gaussian over the training set. This bridge is what makes the results relevant to standard contrastive learning. The paper supports it with (a) a citation to Betser et al. (2026), described as having 'reported' approximate Gaussianity — empirical, not a theorem — and (b) Figures 11–13, showing similar accuracy and filters under both losses. But no theorem guarantees that InfoNCE training induces Gaussian representations, and the paper itself uses hedging language ('approximately Gaussian'). If y(x) is significantly non-Gaussian, the GUPA optimum (partial whitening of DFT power) need not be the InfoNCE optimum, and the theoretical predictions would not transfer to standard CL. This is the single point where the argument is least secure: the mathematical results are about GUPA, but the motivation and experiments are about InfoNCE, and the equivalence rests on an unproven distributional assumption. A secondary concern (also noted by the reader) is that Theorem 3.2's proof assumes cyclic crop plus noise where η is independent of x (Appendix F, Eq. 37), while real crop translations yield η(t) = x(t−Δ)(W(t)−W(t−Δ)) which is explicitly x-dependent. This could break the diagonality of B for real crops, but the paper is transparent about this approximation and provides empirical validation (Figure 14). The Gaussianity concern is more load-bearing because it affects the entire theoretical framework's connection to practice, not just one augmentation type.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This paper analytically computes the optimal representation for contrastive learning (CL) under a proposed loss, LGUPA, for datasets with stationary statistics and a range of standard augmentations (circular crop, linear jitter, ideal blur, crop-plus-noise). The authors show that the globally optimal representation under LGUPA performs partial whitening of DFT power coefficients, computable by a single-layer CNN with sinusoidal filters, a pointwise nonlinearity, global average pooling, and a linear projection. The optimal frequencies and weights are determined by a waterfilling algorithm on the dataset's expected power spectrum. Experiments on synthetic noise, dead leaves, and real image datasets (CIFAR10/100, ImageNet) show that CNNs trained with SGD learn sinusoidal filters and perform partial whitening, consistent with the theoretical predictions.","tokens_in":23950,"tokens_out":1096,"duration_ms":248463,"significance":"The paper provides a valuable and analytically tractable theory connecting contrastive learning to classical natural image statistics. The derivation of the waterfilling optimality (Theorems 1.3, 3.1, 3.2) from first principles—stationarity, the CLT for DFT coefficients (Theorem 2.1), and the diagonal structure of the alignment and covariance matrices—is mathematically sound and yields falsifiable, parameter-light predictions (given the power spectrum). The empirical confirmation that SGD-trained CNNs converge to the predicted sinusoidal filters and partial whitening sensitivities across diverse datasets and augmentations is a notable strength. The bridge from LGUPA to InfoNCE via the Gaussianity assumption is the primary point of fragility, but the authors are transparent about this limitation.","major_comments":[{"comment":"Theorem 1.1 equates the InfoNCE loss to the LGUPA loss only when the representation y(x) is Gaussian over the training set. This is the linchpin connecting the theoretical results (which are about LGUPA) to standard contrastive learning practice (which uses InfoNCE). The paper supports this bridge with a citation to Betser et al. (2026), described as having 'reported' approximate Gaussianity (empirical, not a theorem), and with Figures 11-13 showing similar accuracy and filters under both losses. However, no theorem guarantees that InfoNCE training induces Gaussian representations, and the paper uses hedging language ('approximately Gaussian'). If y(x) is significantly non-Gaussian, the GUPA optimum (partial whitening of DFT power) need not be the InfoNCE optimum. The authors should strengthen this connection: for example, by providing a direct quantitative comparison of the two losses (","section":null},{"comment":"Theorem 3.2's proof (Appendix F, Eqs. 36-37) models crop translations as cyclic translations plus noise, where the noise term η(t) = x(t−Δ)(W(t)−W(t−Δ)) is treated as independent of x. As the paper acknowledges, η(t) is explicitly signal-dependent. The diagonal structure of the B matrix (Appendix G) is derived under the assumption that η is independent white Gaussian noise. While Figure 14 provides empirical evidence that the two augmentation types behave similarly in the DFT domain, the signal-dependence of η means the B matrix for real crop translations may not be exactly diagonal, potentially affecting the optimality of sinusoidal filters. The authors should discuss the sensitivity of the theoretical predictions to this signal-dependence, or clarify the scope of Theorem 3.2 as applying strictly to the cyclic-crop-plus-independent-noise model.","section":null}],"minor_comments":[{"comment":"In Algorithm 1, the loop runs 'for i = 1 to 1/η'. It would be clearer to specify the stopping criterion (e.g., until total power budget is exhausted or convergence).","section":null},{"comment":"Figure 3 references '32 randomly chosen squared DFT coefficients' but the covariance matrix shown appears to be larger. Clarify the dimensionality.","section":null},{"comment":"The paper states (Section 4) that σ and temperature t were fixed (σ=0.0014, t=1) for all experiments. A brief discussion of how robust the predicted ring/diamond sensitivity patterns are to these hyperparameters would be helpful, since the waterfilling solution depends on them.","section":null},{"comment":"The reference to 'Betser et al., 2026' appears to be a future-dated citation. If this is a forthcoming publication, please verify the citation details.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The Gaussianity bridge (Theorem 1.1) is the weakest link, but the authors are admirably transparent about it in Section 5. The core mathematical contributions (Theorems 2.1, 3.1, 3.2) are sound for the LGUPA loss, and the empirical validation is thorough. I recommend minor revision with a request to strengthen the discussion of the Gaussianity assumption's robustness and the signal-dependent noise approximation. The paper is a good fit for the journal."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. The two major comments both concern the scope of our theoretical guarantees relative to standard contrastive learning practice. We address each below and describe revisions we will make.","responses":[{"response":"We agree that the Gaussianity bridge is the primary point of fragility in our theory, and we appreciate the referee's precise articulation of the issue. We want to be clear about what we can and cannot claim. What we can claim: (1) Theorem 1.1 is a conditional result—if y is Gaussian, then InfoNCE and LGUPA are equivalent up to a constant. This is a theorem, not an approximation. (2) Empirically, representations trained with InfoNCE are approximately Gaussian (Betser et al., 2026), and training with LGUPA versus InfoNCE yields nearly identical learned filters (Figure 12) and nearly identical recognition accuracy across multiple augmentations and architectures (Figures 11, 13). (3) The same partial-whitening structure arises under other contrastive losses that do not require Gaussianity, such as VICReg (Bardes et al., 2022) and Barlow Twins (Zbontar et al., 2021), which are based on covariance rather than full distributional assumptions. This suggests the predictions are not an artifact of the Gaussian assumption specifically. What we cannot claim: we have no theorem guaranteeing that InfoNCE training induces Gaussian representations, nor can we rule out that for some datasets or architectures, significant non-Gaussianity could cause the LGUPA optimum to diverge from the InfoNCE optimum. We will revise the manuscript to make the scope of Theorem 1.1 more explicit and to add a quantitative comparison of the two losses. Specifically, we will add a table reporting the KL divergence (or other distributional distance) between the learned representation distribution and a fitted Gaussian for each dataset and augmentation, so readers can assess the degree of Gaussianity directly. We will also add a brief discussion of why the predictions may be robust to moderate non-Gaussianit","revision_made":"partial","referee_comment":"Theorem 1.1 equates InfoNCE to LGUPA only when y(x) is Gaussian. This is the linchpin connecting theory to practice. No theorem guarantees InfoNCE induces Gaussian representations. If y is significantly non-Gaussian, the GUPA optimum need not be the InfoNCE optimum. Strengthen this connection."},{"response":"The referee is correct that η(t) is signal-dependent and that this means the B matrix for actual crop translations is not guaranteed to be exactly diagonal. Our current treatment in Appendix F acknowledges this signal-dependence but does not rigorously bound its effect on the diagonal structure of B. We will make two changes. First, we will clarify the scope of Theorem 3.2 (and its appendix version, Theorem G.1) as applying strictly to the cyclic-crop-plus-independent-Gaussian-noise model. This is already implicit in the proof but should be stated explicitly in the theorem statement. Second, we will add a discussion of the sensitivity to the signal-dependence. The key observation is that the diagonal structure of B requires not that η be independent of x, but rather that the cross-frequency terms E[δ[k]δ[l]] vanish for k≠l. For the signal-dependent noise model, these cross terms involve E[|xF[k]|² · (terms involving ηF)] which, under the Gaussianity of DFT coefficients (Theorem 2.1), factor into products of expectations. We will verify empirically that the off-diagonal elements of B for actual crop translations are small relative to the diagonal, complementing the visual evidence in Figure 14. We agree that a full theoretical treatment of the signal-dependent case is beyond the scope of this paper, and we will state this as an open problem. We believe the empirical evidence (Figure 14 showing similar DFT-domain behavior, and the experimental results in Figures 7-8 showing sinusoidal filters for crop translations across all datasets) supports the practical relevance of the theory, but we will be more precise about what is proven versus what is empirically supported.","revision_made":"partial","referee_comment":"Theorem 3.2's proof models crop translations as cyclic translations plus noise, where η(t) = x(t−Δ)(W(t)−W(t−Δ)) is treated as independent of x. The noise is explicitly signal-dependent. The diagonal structure of B is derived under the assumption that η is independent white Gaussian noise. The B matrix for real crop translations may not be exactly diagonal. Discuss sensitivity or clarify scope."}],"tokens_in":23416,"tokens_out":1308,"duration_ms":66449,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper gives a genuine analytical result connecting contrastive learning to classical natural image statistics. The authors show that for stationary image datasets and a range of augmentations, the global optimum of their GUPA contrastive loss is partial whitening of DFT power coefficients, computable by a single-layer CNN with sinusoidal filters and a waterfilling algorithm on the dataset's power spectrum. That is a real contribution — the connection from stationarity to DFT independence (via Peligrad & Wu's CLT) to diagonal B and Σ matrices to waterfilling is clean and the proofs check out. The experiments confirm the predictions: trained CNNs do learn sinusoids and partial whitening across multiple datasets and augmentations. The authors are also honest about limitations, which I appreciate. Credit is earned here. The core mathematical chain is sound and the experiments are directly motivated by the theory rather than bolted on. The waterfilling derivation and the extension to ReLU nonlinearities (Theorem E.1) are nontrivial. The one load-bearing soft spot is Theorem 1.1, which equates GUPA to InfoNCE only when the representation y(x) is Gaussian over the training set. This bridge is what makes the theory relevant to standard contrastive learning. The paper supports it with a citation to Betser et al. (2026) — empirical, not a theorem — and Figures 11–13 showing similar accuracy and filters under both losses. That is reasonable empirical evidence but it is not a proof. If y(x) is significantly non-Gaussian, the GUPA optimum need not be the InfoNCE optimum. The stress-test note flags this correctly. I would add: the paper could have strengthened its case cheaply by directly comparing the two losses for the predicted optimal representation rather than only comparing trained networks. A secondary concern: Theorem 3.2's proof models crop translations as cyclic translation plus noise where the perturbation η is independent of x, but real crops give η(t) = x(t−Δ)(W(t)−W(t−Δ)), which is explicitly x-dependent. The paper is transparent about this and Figure 14 provides empirical validation. This is a minor issue — the approximation is reasonable and the paper does not oversell it. No code is provided, which is a practical gap but not a theoretical one. Who is this for? Researchers in representation learning theory and anyone interested in why contrastive learning works with simple augmentations. The reader who wants to understand the mathematical structure of contrastive losses on stationary signals gets real value. It does not solve deep networks or SOTA augmentations, and it does not claim to. This deserves a serious referee. The Gaussianity question is the thing to press on in review — whether a theorem can be proven or whether stronger empirical evidence can be demanded — but the paper's core contribution stands on its own merits.","headline":"The Gaussianity bridge between GUPA and InfoNCE is the one real soft spot in an otherwise clean and genuinely novel theory paper.","tokens_in":24577,"tokens_out":662,"would_cite":true,"duration_ms":161219,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Contrastive learning on images reduces to partial whitening of Fourier power","keywords":["contrastive learning","partial whitening","waterfilling algorithm","stationary image statistics","DFT coefficients","sinusoidal filters","InfoNCE loss","natural image power spectrum"],"falsifier":"Train a contrastive model on a dataset with strongly non-Gaussian DFT coefficients (violating stationarity) and check whether the learned filters deviate from sinusoids and the partial-whitening sensitivity pattern — if they do, the stationarity assumption is load-bearing rather than incidental.","tokens_in":23708,"feed_emoji":"🌊","tokens_out":1459,"duration_ms":159715,"temperature":0.7,"pith_summary":"This paper asks why contrastive learning produces useful image representations from simple augmentations and simple images. The authors analytically compute the globally optimal representation under a contrastive loss (called GUPA, which equals the standard InfoNCE loss when representations are Gaussian) for any image dataset with stationary statistics. They prove that for a range of augmentations — circular crop, linear jitter, ideal blur, and crop-plus-noise with arbitrary blur — the optimal representation is partial whitening of the discrete Fourier transform's power coefficients: select a subset of spatial frequencies, measure their power, and rescale each so that the representation has equal variance in all directions. The specific frequencies and their weights are determined by a waterfilling algorithm on the dataset's expected power spectrum. This optimal representation can be computed by a single-layer CNN whose first-layer filters are sinusoids, followed by a pointwise nonlinearity (squaring or ReLU), global average pooling, and a linear projection. The key mechanism is that stationarity of the image statistics forces DFT coefficients to be asymptotically Gaussian and pairwise independent, which diagonalizes both the covariance and augmentation-distortion matrices in Fourier space, making the optimization separable across frequencies. Experiments confirm that CNNs trained by SGD on real and synthetic datasets learn sinusoidal first-layer filters and perform partial whitening, matching the theoretical predictions.","feed_headline":"Optimal contrastive representation is partial whitening of Fourier power","feed_subtitle":"For any stationary image dataset, the best contrastive loss solution selects frequencies by waterfilling and rescales them — a single-layer ","key_machinery":"stationarity → Gaussian independent DFT coefficients → diagonal loss in Fourier space → waterfilling selects frequencies → partial whitening rescales them","core_discovery":"The central object is the partial whitening representation: for stationary image datasets, the contrastive loss is globally minimized by measuring power at a subset of spatial frequencies (selected by a waterfilling algorithm on the expected power spectrum) and rescaling each frequency's power inversely to its expected variance. The stationarity of natural images — meaning their statistics are translation-invariant — is the load-bearing property that makes this work, because it forces DFT coefficients to be Gaussian and pairwise independent, which in turn makes the contrastive loss separable across individual frequencies. A single-layer CNN with sinusoidal filters, a pointwise nonlinearity,,","pith_inferences":["The gap between the simple-augmentation regime (where partial whitening is optimal) and SOTA augmentations (where localized filters emerge) suggests that standard augmentations like randomized crop size and nonlinear color jitter break the frequency-separability that makes the analysis tractable — extending the theory would require handling non-diagonal augmentation-distortion matrices in Fourier ","If the Gaussianity assumption on representations fails for certain architectures or datasets, the GUPA loss diverges from InfoNCE and the optimality guarantees may not transfer; however, the empirical observation that learned representations are approximately Gaussian suggests this may be a mild restriction in practice.","The non-uniqueness of the optimal solution (any orthogonal rotation of the whitened representation is also optimal) means that gradient descent dynamics, not the loss landscape alone, determine which specific frequencies are learned — connecting implicit bias of optimization to the observed filter structure.","The finding that most recognition accuracy gains come from partial whitening rather than learned invariance suggests that simple spectral preprocessing might recover much of the benefit of contrastive pretraining for downstream tasks, at least for simple architectures."],"forward_implications":["If the theory extends to deeper architectures, it would explain why early layers of contrastive learning models converge to Gabor-like or sinusoidal filters: they are computing the analytically optimal partial whitening representation dictated by the dataset's power spectrum.","The waterfilling algorithm provides a parameter-free prediction of which spatial frequencies a contrastive model should be sensitive to, given only the dataset's expected power spectrum — this is directly testable on any dataset and augmentation set.","The result that synthetic noise images transfer to real images only when their power spectra match explains the empirical finding that 1/f noise works better than white noise for pretraining: the waterfilling solution selects the same frequencies only when the spectra align.","The theory predicts that augmentations do not teach invariance per se but instead bias the waterfilling solution toward particular frequency bands, which means augmentation design can be reframed as frequency-band selection rather than invariance engineering."],"fun_headline_variants":["Contrastive loss for natural images minimizes via partial Fourier whitening","Optimal contrastive representations use waterfilling and partial Fourier whitening","Stationary images make optimal contrastive representations partial Fourier whitening","Waterfilling on the power spectrum defines optimal contrastive representations","Optimal contrastive CNNs learn sinusoids to perform partial frequency whitening"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The theoretical results assume that the learned representation is Gaussian-distributed over the training set, which is what makes the analytically tractable GUPA loss equal to the standard InfoNCE contrastive loss. If the representation is not Gaussian, the optimality guarantees may not apply to standard contrastive learning as practiced.","fun_headline_variants_meta":{"raw":{"variants":["Contrastive loss for natural images minimizes via partial Fourier whitening","Optimal contrastive representations use waterfilling and partial Fourier whitening","Stationary images make optimal contrastive representations partial Fourier whitening","Waterfilling on the power spectrum defines optimal contrastive representations","Optimal contrastive CNNs learn sinusoids to perform partial frequency whitening"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":1417,"prompt_tokens":464,"completion_tokens":953,"prompt_tokens_details":null},"tokens_in":464,"tokens_out":953,"duration_ms":111842,"temperature":1.0,"reasoning_tokens":919,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T09:46:31.918796+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Train a contrastive model on a dataset with strongly non-Gaussian DFT coefficients (violating stationarity) and check whether the learned filters deviate from sinusoids and the partial-whitening sensitivity pattern — if they do, the stationarity assumption is load-bearing rather than incidental.","supporting_citations":[],"review_version":1}