{"id":"7d0a53f7-f5c9-466a-830b-5b7e0b0bb834","arxiv_id":"2502.10236","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Frequency-filtered noise in the diffusion forward process steers what the denoiser learns, yielding modest FID gains on some datasets and partial recovery after known-band corruption.","lead":"This paper proposes replacing the white noise used in diffusion model training with Gaussian noise that is filtered in the frequency domain, so the model can be steered to learn or ignore specific image frequencies. On several datasets the schedule improves FID, and a corruption-aware variant produces samples closer to the original distribution than standard diffusion.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Frequency-shaped noise breaks the standard DDPM training/sampling assumptions, so the reported FID gains may reflect a heuristic objective rather than a validated inductive-bias mechanism.","rationale":"The reader's weakest assumption correctly identifies the missing derivation of the reverse process and loss weighting when the noise is colored. I examined Section 2.2 and found no derivation of the forward marginal, the posterior, or the initialization prior; Eq. (3) is written for white noise, and Eq. (7) is inserted without comment. The covariance of epsilon^(w) is not identity for non-constant w, so the standard ELBO derivation in Ho et al. (2020) does not carry over. This is load-bearing because the paper's headline claims about steering inductive bias and improved performance are empirical comparisons that presuppose the modified process is a valid diffusion model. If the objective is only a heuristic, the FID improvements could be due to a different effective loss (e.g., a reweighted denoising objective) rather than the frequency content. I agree with the reader that this places the central conclusions on weak footing. The paper's controlled experiments and use of multiple seeds are a positive sign, but they cannot compensate for the missing theoretical consistency. The proposed analytic check would directly test the validity of the sampling procedure, and the corrected-objective comparison would test whether the empirical gains survive a principled formulation.","tokens_in":11145,"tokens_out":9528,"duration_ms":103500,"concrete_test":"Derive the exact reverse posterior for the forward process x_t = sqrt(bar_alpha_t)x_0 + sqrt(1-bar_alpha_t)epsilon^(w) with epsilon^(w) ~ N(0,Sigma_w), and compare it with the standard DDPM posterior used in sampling. On a Gaussian target with known covariance, compute the KL divergence between the true posterior and the paper's reverse step; if nonzero, the sampler does not target the data distribution. This analytic check directly settles whether the missing derivations in Section 2.2 invalidate the training and sampling claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central premise is that epsilon^(w) from Eq. (7) can be substituted into the standard DDPM forward process (Eq. 4) and trained with the unit-variance epsilon-prediction loss (Eq. 3) without changing the training or sampling machinery. This is not justified. epsilon^(w) is Gaussian with non-identity covariance Sigma_w = F^{-1} diag(|w(f)|^2) F (up to FFT normalization), so the forward marginal and the reverse posterior q(x_{t-1}|x_t,x_0) are no longer those of standard DDPM. The ELBO for this forward process would require a covariance-aware loss (inverse Sigma_w weighting) and a modified reverse covariance; the paper derives neither. Nor does it specify the FFT normalization or the prior used at sampling time, both of which change the noise variance and hence the effective signal-to-noise ratio. If the unmodified objective and sampler are used, the training is not maximizing a valid ELBO for the stated generative process, and the FID comparisons in Tables 1 and 2 may be between a correctly trained baseline and a heuristically trained variant. The claimed frequency steering could then be an artifact of an under-specified objective rather than the inductive-bias mechanism.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes 'frequency diffusion', a modification of the DDPM forward process in which isotropic Gaussian noise is replaced by frequency-shaped Gaussian noise produced by filtering complex Gaussian noise in the Fourier domain and taking the real part. The authors argue that this noising operator acts as an inductive-bias knob: erasing certain frequency bands creates pressure on the denoiser to learn those bands, while leaving other bands intact lets the model ignore them. They define two-band mixture schedules (Eq. 12), train diffusion models on MNIST, CIFAR-10, DomainNet-Quickdraw, WikiArt, and CelebA, and report FID/KID comparisons against standard diffusion. They also present a corruption-recovery experiment where data corrupted in a known frequency band is used to train a model that, with a frequency-omitting schedule, approximately recovers the original uncorrupted distribution.","tokens_in":11394,"tokens_out":4300,"duration_ms":48825,"significance":"The underlying idea is attractive: if the forward noising operator can be used to steer what the denoising model learns, the forward process becomes a meaningful design choice for generative priors, and the qualitative figures (Figure 2 and Figure 4) support the intuition. The paper also deserves credit for stating its limitation explicitly: Section 4 acknowledges that schedule design currently requires empirical validation. However, the central empirical claim is not yet supported by the reported numbers, and the theoretical foundation is incomplete: substituting non-white noise into a standard DDPM objective without re-deriving the loss or the sampling procedure leaves the comparisons in Tables 1 and 2 on questionable footing. The corruption-recovery result is a useful proof-of-concept but is constructed from knowledge of the corruption band, which limits its scope. Overall, the contribution is promising but not yet at a level where the main claims are established.","major_comments":[{"comment":"The paper replaces epsilon ~ N(0,I) with epsilon^(w) = Re(F^{-1}(w ⊙ N_freq)) but keeps the standard epsilon-prediction loss (Eq. 3) and the standard forward scaling (Eq. 4) without deriving the correct training or sampling objective. epsilon^(w) is Gaussian with non-identity covariance: its power spectrum is |w(f)|^2, so the forward marginal and the reverse posterior q(x_{t-1}|x_t,x_0) are not those of standard DDPM. A valid ELBO would require a covariance-aware loss (inverse-covariance weighting) and a modified reverse covariance; the paper provides neither. It also does not specify the FFT normalization or the distribution used to initialize sampling, both of which affect the effective signal-to-noise ratio. The statement in Section 1 that the Fourier transform of a Gaussian is Gaussian only preserves Gaussianity, not the identity covariance that the standard objective relies on. This must be fixed or explicitly justified before the FID comparisons can be interpreted as comparing validly trained models.","section":"Section 2.2, Eq. (7)"},{"comment":"The claim that frequency diffusion 'outperforms standard diffusion' on three of five datasets is not supported by the reported numbers when standard errors are considered. For example, on CIFAR-10 the baseline FID is 0.1055±0.0042 while the best reported settings (gamma_l=0.4 and 0.6) give 0.1131±0.0079 and 0.1131±0.0074, which are worse than the baseline. On DomainNet-Quickdraw the baseline is 0.0875±0.0060 and the best is 0.0799±0.0166, a gap that is well within one standard error. On WikiArt the baseline 0.1622±0.0133 overlaps with the best 0.1348±0.0126, and no significance test is reported. Furthermore, the best gamma_l per dataset is selected by sweeping on the test FID, making the reported best-vs-baseline gap a post-hoc best-of-nine result, not an independent prediction. The paper should either report held-out model selection, provide significance tests or confidence intervals, or substantially weaken the performance claims.","section":"Section 3.1.3, Table 1"},{"comment":"The corruption-recovery experiment is constructed from the known corruption band: the authors set b_l = a_c and a_h = b_c so that the forward noising operator omits exactly the frequency range in which the corruption noise lies. This demonstrates that a schedule can be designed when the corruption is fully known, but it is not a recovery method for unknown corruption, and the framing in Section 3.2 as 'recovery of the original, noiseless, distribution' overstates the result. The paper should explicitly state that the schedule is hand-designed from knowledge of the corruption process, and it should present the experiment as a proof-of-concept of the omission mechanism rather than as a general recovery result.","section":"Section 3.2, Eq. (13)"}],"minor_comments":[{"comment":"The MNIST dataset is cited to Matthey et al. (2017), which is the dsprites dataset paper; the correct reference for MNIST is LeCun et al. (1998) or a standard source.","section":"Section 2.4, References"},{"comment":"There are several typos: 'nosing' should be 'noising', 'stong' should be 'strong', and 'DPS' should be 'DPMs'.","section":"Section 4, Discussion"},{"comment":"The two-band mixture is introduced as epsilon_f = gamma_l epsilon_[a_l,b_l] + gamma_h epsilon_[a_h,b_h], but the paper does not state whether the two band-limited noises on the right are independent or whether the gamma weights are normalized to preserve total noise variance; this matters for the claim that gamma_l = 0.5, gamma_h = 0.5 yields 'standard Gaussian noise'.","section":"Section 2.3, Eq. (12)"},{"comment":"The text says the average FID and error are reported in Figure 3, but the caption does not describe how error bars or shaded regions are computed; please clarify the visualization.","section":"Section 3.1.2, Figure 3"}],"recommendation":"major_revision","confidential_remarks":"This is a workshop-length paper, and my recommendation is calibrated to that scope. The idea is interesting and the qualitative demonstrations are suggestive, but the manuscript currently lacks the theoretical derivation needed to make the training objective valid and the empirical evidence does not yet support the performance claims. Both issues are fixable in principle: the authors could derive the covariance-aware objective (or show invariance), and they could report a proper model-selection procedure with significance-aware metrics. If those changes alter the conclusions, the paper would need another round of review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead this one with a cup of coffee. The corruption-recovery experiment is genuinely useful: if you know a band of corruption, you can omit it from the forward noise and train a diffusion model that largely ignores it. The FID gap in Table 2 is real and that result is worth building on.\n\nThe natural-dataset story does not hold up. Table 1 shows baseline or tied-best for MNIST and CIFAR-10; the best gamma variant beats baseline by less than two standard errors on the other three. Calling that 'significantly benefit' is an overstatement, especially since the winning gamma was chosen post-hoc from a nine-point sweep on test FID. That is cherry-picking unless they hold out a validation split and test the chosen schedule once.\n\nThe stress-test note is on target about the theory. epsilon^(w) has non-identity covariance, so the standard epsilon-prediction loss is not the ELBO for that forward process. No covariance-aware weighting, posterior variance, or sampling prior is provided. The claim as written that gamma_l=0.5, gamma_h=0.5 gives white noise also looks wrong without a normalization step; as written, the variance would not be one. This makes the method a heuristic as presented, not a derived algorithm. That is not fatal if the empirical side is solid, but here the empirical side is only solid for the corruption case.\n\nCredit where due: the idea is clearly motivated, the two-band setup is a natural prior, and the corruption-omission protocol is new as far as I know. They report seeds and standard errors, which is good practice. The failure is in the gap between claim and evidence.\n\nWho is this for? People thinking about inductive bias in diffusion or denoising with known corruption. It deserves a serious referee but not publication in current form. A referee should ask for the derivation of the true objective, a validation-based schedule selection, and significance tests on the table.\n\nRecommendation: accept to peer review with heavy revision expected. I would not cite it as evidence of improved FID until the numbers are redone.","headline":"The corruption-recovery experiment is a real result; the FID gains on natural datasets and the theory are too thin for the headline claims.","tokens_in":11908,"tokens_out":7645,"would_cite":false,"duration_ms":79327,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that the noising process in a diffusion model is not just a computational convenience but a direct lever on what the model learns: whatever information the forward process erases is exactly what the denoiser must…","keywords":["diffusion probabilistic models","frequency-domain noise","inductive bias","noise schedule","colored Gaussian noise","generative modeling","FID","spectral bias"],"falsifier":"Derive the exact marginal $q(x_t|x_0)$ for the frequency-weighted forward process; if the conditional distribution of the noise target is not the zero-mean unit-variance Gaussian assumed in the $\\epsilon$-prediction loss, then the loss in Eq. (3) is no longer a valid bound and the FID gains could come from a heuristic rather than from inductive-bias steering. A direct experiment would train with frequency-shaped forward noise but sample the reverse process with standard white noise at every step, and vice versa; if matching the noise spectrum does not change sample quality, the claim that the noising operator steers learning is falsified.","tokens_in":10933,"feed_emoji":"🎛️","tokens_out":9192,"duration_ms":83926,"temperature":0.7,"pith_summary":"This paper argues that the noising process in a diffusion model is not just a computational convenience but a direct lever on what the model learns: whatever information the forward process erases is exactly what the denoiser must reconstruct, so choosing which frequencies get erased steers the model's inductive bias. To test this, the authors replace the usual white Gaussian noise with Gaussian noise whose power spectrum is shaped by a frequency weighting, calling the result frequency diffusion. They report that on data whose information sits mainly in low frequencies, low-frequency-weighted forward noise gives better FID than standard diffusion, and across five datasets different frequency weightings win, with three datasets improving over the Gaussian baseline. They further show that if training data is corrupted with noise restricted to a frequency band, training with a forward noising operator that leaves that band untouched lets the model ignore the corruption and generate samples closer to the original uncorrupted distribution. The paper positions the forward noising operator as a tunable inductive-bias knob for diffusion models.","feed_headline":"The noise you add decides what diffusion models learn","feed_subtitle":"Gaussian noise shaped in frequency space lets diffusion models focus on, or ignore, specific image details.","key_machinery":"The machinery is the frequency-controlled noise operator, which turns standard white Gaussian noise into Gaussian noise with a chosen power spectrum. Starting from a complex Gaussian field in Fourier space, the operator scales each frequency component by a weighting $w(f_x, f_y)$ and returns $\\epsilon^{(w)} = \\Re(\\mathcal{F}^{-1}(N_{\\mathrm{freq}}^{(w)}))$, a real-valued spatial noise field. Because linear operations on Gaussians produce Gaussians, the training loop keeps the usual $\\epsilon$-prediction objective while the noise's frequency content acts as the inductive-bias knob. The experiments use a two-band mixture of low- and high-frequency filtered noises with mixing weights $\\gamma_l$ and $\\gamma_h$, and the corruption-recovery task uses the same construction with the corrupted band excluded from the forward process.","core_discovery":"The paper's central claim is that the noising operator directly shapes a diffusion model's representation of the data, because the denoising network is under pressure to learn precisely the information that the forward process destroys. In the Fourier domain, the authors build frequency-shaped Gaussian noise by sampling a complex Gaussian field $N_{\\mathrm{freq}} = N_{\\mathrm{real}} + i N_{\\mathrm{imag}}$, multiplying each frequency bin by a weighting function $w(f_x,f_y)$, and inverse-transforming to the spatial domain to get $\\epsilon^{(w)} = \\Re(\\mathcal{F}^{-1}(N_{\\mathrm{freq}}^{(w)}))$; standard white Gaussian noise is the special case $w(f) = 1$. Using a two-band mixture $\\epsilon_f = \\gamma_l \\epsilon_{[a_l,b_l]} + \\gamma_h \\epsilon_{[a_h,b_h]}$, the paper sweeps the low/high balance and finds dataset-dependent, often monotonic, FID rankings: low-frequency-forward schedules win on low-frequency-dominated data, while high-frequency-forward schedules win on datasets where sharp details matter. In the corruption experiments, when the data is degraded by band-restricted noise $\\epsilon_{[a_c,b_c]}$, setting the forward noising operator to avoid that band removes the denoiser's pressure to learn the corruption, and the trained model produces samples closer to the original uncorrupted distribution than standard diffusion in all eight tested bands. The paper reads these results as evidence that frequency-based noise control is a practical way to inject task-specific inductive biases into diffusion training.","pith_inferences":["If the noising operator is a true inductive-bias knob, dynamic schedules that shift frequency emphasis over diffusion time—coarse structure first, fine texture later—should outperform any constant weighting; this is an untested extension the authors mention as future work.","The corruption-recovery result likely extends beyond images to any domain with a meaningful Fourier structure, such as audio, time series, or physics fields, where band-restricted degradations are common.","The deepest implication is that any structured degradation expressible as a subspace of a Gaussian noise process could in principle be omitted from the forward process, making 'ignore the corrupted part of the signal' a general recipe rather than a frequency-specific trick.","A cheap practical consequence: because the method requires no architecture change, frequency diffusion is a natural baseline for future inductive-bias work in generative modeling."],"forward_implications":["The noising operator becomes a tunable inductive-bias knob: dataset-specific frequency schedules can match the dominant information content, and the paper reports lower FID and KID than the white-noise baseline on three of five tested datasets.","Frequency-shaped noise enables selective learning: a model trained on band-corrupted data with the corrupted band omitted from the forward process generates samples closer to the original uncorrupted distribution than a standard baseline, across all eight tested bands.","Monotonic FID trends across $\\gamma_l$ sweeps can reveal which frequency bands carry the information needed for generation, making the noise spectrum a diagnostic tool rather than a fixed hyperparameter.","The framework strictly generalizes standard diffusion, since white Gaussian noise is recovered as the $w(f)=1$ special case, so no architectural change is needed to apply the method."],"supporting_citations":[{"why":"supplies the DDPM epsilon-prediction objective and sampling procedure that frequency diffusion modifies.","marker":"Ho et al., 2020a"},{"why":"establishes the forward-reverse diffusion formalism whose noising operator the paper manipulates.","marker":"Sohl-Dickstein et al., 2015"},{"why":"shows that denoising networks' inductive biases drive diffusion generalization, the prior result the paper extends to the noising side.","marker":"Kadkhodaie et al., 2023"},{"why":"studies inductive and primacy biases in diffusion models, motivating the attempt to shape bias explicitly.","marker":"Zhang et al., 2024"},{"why":"demonstrates diffusion with non-Gaussian corruption, a precedent for replacing the standard noising operator.","marker":"Bansal et al., 2022"},{"why":"shows the noise schedule itself can be learned and tuned, supporting the idea that the forward process is a design choice.","marker":"Sahoo et al., 2024"},{"why":"documents the spectral bias of neural networks, the frequency-learning phenomenon the paper's inductive-bias argument relies on.","marker":"Rahaman et al., 2019"}],"fun_headline_variants":["Noise frequency shapes diffusion's inductive bias","Frequency-tuned noise steers what diffusion learns","Diffusion's focus is set by noise frequency","Shaping noise in frequency space biases diffusion","Noise's frequency control biases diffusion training"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The comparison rests on the assumption that replacing white Gaussian noise with frequency-filtered Gaussian noise in the forward process still leaves the standard diffusion training loss valid; the paper does not derive the reverse-process covariance, adjust the loss weighting, or specify what noise is added during sampling for the colored case.","fun_headline_variants_meta":{"raw":{"variants":["Noise frequency shapes diffusion's inductive bias","Frequency-tuned noise steers what diffusion learns","Diffusion's focus is set by noise frequency","Shaping noise in frequency space biases diffusion","Noise's frequency control biases diffusion training"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00021,"raw_usage":{"total_tokens":1445,"prompt_tokens":1013,"completion_tokens":432,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":629,"completion_tokens_details":{"reasoning_tokens":364}},"tokens_in":629,"tokens_out":432,"duration_ms":4966,"temperature":1.0,"reasoning_tokens":364,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T18:51:34.949282+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Derive the exact marginal $q(x_t|x_0)$ for the frequency-weighted forward process; if the conditional distribution of the noise target is not the zero-mean unit-variance Gaussian assumed in the $\\epsilon$-prediction loss, then the loss in Eq. (3) is no longer a valid bound and the FID gains could come from a heuristic rather than from inductive-bias steering. A direct experiment would train with frequency-shaped forward noise but sample the reverse process with standard white noise at every step, and vice versa; if matching the noise spectrum does not change sample quality, the claim that the noising operator steers learning is falsified.","supporting_citations":[],"review_version":1}