{"id":"76e01659-38df-4656-b1b0-2e75049e18a3","arxiv_id":"2504.19353","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"K-Flow trains flow-matching models with frequency scale as time, enabling competitive image and molecule generation plus scale-level control of outputs.","lead":"K-Flow turns image generation into a journey through frequencies, starting with coarse shapes and adding fine details as a scale knob increases. It is worth reading because it offers a way to steer generated images, keeping some details fixed while changing others, on top of ordinary flow matching.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (10) does not define the claimed interpolant: the noise term (1−I_{r≥⌊k⌋+1})ε is active on bands already set to data or interpolation, so Ψ_k≠φ_k at integer k and Eq. (11)/(12) do not follow.","rationale":"The reader's weakest_assumption identifies precisely the load-bearing concern: Eq. (10) is not a valid stochastic interpolant because the noise-padding term is active on bands that are already occupied by data or interpolation. I agree with that assessment. The central claim of the paper—that K-Flow enables flow matching across the scaling parameter as time—depends on the path Ψ_k being a genuine noise-to-data interpolation and on Eq. (11) being its derivative. As written, Eq. (10) fails at integer k, the one regime where the discrete construction is unambiguous. The manuscript itself contains an internal contradiction: the bullet list below Eq. (10) describes the noise term as I_{r≥⌊k⌋+1}ε, but the displayed equation uses its complement. The empirical results cannot compensate for this because the training target in Eq. (12) is the derivative of the incorrect path. Additional secondary issues—no released code or training details, the conclusion claiming verification exclusively on images despite the molecule-assembly experiments, and the unsupported VAR-as-special-case statement—reinforce the need for a corrected derivation and reproducible implementation, but they are not the primary reason for rejection. Since the identified defect is load-bearing and the reader already reached REJECT, no verdict adjustment is needed; the recommendation remains that the paper should not be accepted in its current form.","tokens_in":20540,"tokens_out":6689,"duration_ms":68142,"concrete_test":"Evaluate Eq. (10) at a single integer step for a 1D signal, e.g. N=4 Fourier coefficients, at k=1, t=0. Compute the coefficient at r=0: the first term gives Fφ(0) and the second term gives ε(0), so Ψ_1(0)=Fφ(0)+ε(0), whereas Eq. (9) gives φ_1(0)=Fφ(0). If the two differ, Eq. (10) does not interpolate the discrete flow and Eq. (11)/(12) do not follow. Equivalently, replace the second term with I_{r≥⌊k⌋+1}ε and recheck both endpoint conditions μ(0)=0, μ(1)=1; if the corrected interpolant satisfies Ψ_k=φ_k at all integer k and Eq. (11) matches its derivative, the central construction is repaired.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive defect is in the K-amplitude interpolant, Eq. (10). Writing r = sqrt(kx^2+ky^2+kz^2), the second term is (1 − I_{r≥⌊k⌋+1})ε, which adds ε to every coefficient with r < ⌊k⌋+1. Consequently: (i) coefficients already set to Fφ by the first term receive Fφ+ε, not Fφ; (ii) coefficients in the active band receive μFφ + (2−μ)ε, not μFφ + (1−μ)ε; (iii) at integer k (t=0, μ(0)=1), the active band equals Fφ+ε, so Ψ_k ≠ φ_k, contradicting the construction and the stated endpoint limit. The bullet list immediately below Eq. (10) describes the intended term as I_{r≥⌊k⌋+1}ε, so the equation and its own explanation disagree. Eq. (11) is therefore not the derivative of Eq. (10); it also reverses the sign of the active-band update (μ′(ε−Fφ) instead of μ′(Fφ−ε)) and drops the contribution of the noise term. Since Eq. (12) trains v_k against this derivative, the conditional-flow-matching objective is attached to a path that is never the discrete K-amplitude flow. Separately, the stated bump conditions μ(0)=μ(1)=1 make the active band data at both endpoints; a noise-to-data interpolation requires μ(0)=0, μ(1)=1. These are internal inconsistencies, not disagreements with prior work. They may be repairable by replacing the second term with I_{r≥⌊k⌋+1}ε and fixing μ, but as submitted the central derivation does not hold.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes K-Flow, a flow-matching generative framework in which the usual generative time variable is replaced by a scaling parameter k indexing frequency-like bands of an invertible K-amplitude transform (Fourier, wavelet, or data-dependent PCA). Section 3.1 defines a continuous interpolant Ψ_k between noise and data, derives a localized conditional vector field dΨ_k/dk, and trains a velocity network with a conditional flow-matching loss. The paper reports FID and Recall results on CelebA-HQ, LSUN Church, and ImageNet, three controllability ablations, and molecular assembly results on COD-Cluster17, concluding that K-Flow achieves competitive or better performance and enables explicit scale-wise steering of generation.","tokens_in":20836,"tokens_out":10195,"duration_ms":106898,"significance":"At a high level, the central idea is attractive and potentially broader than one more diffusion variant: organizing the generation path by the scale index of an invertible transform is a clean conceptual contribution, and the proposed plug-in replacement of the time embedding by a scale embedding is plausible. If the construction in Section 3.1 is corrected, the paper would supply a general recipe for scale-parametrized interpolants, and the qualitative steerability experiments, especially the shared-noise manipulations in Section 5.4, are useful demonstrations. The paper also states its own limitations and mitigation arguments in Section 5.4, which is to its credit. The main difficulty is that the written derivation does not support the method as presented: Equations (10)-(12) contain load-bearing errors, and the empirical tables cannot substitute for a correct central derivation.","major_comments":[{"comment":"The K-amplitude interpolant as written double-counts noise. The second term (1 − I_{r≥⌊k⌋+1})ε is active for every coefficient with r < ⌊k⌋+1, which includes the low band r < ⌊k⌋ that is already set to Fφ in the first term and the active band r∈[⌊k⌋,⌊k⌋+1) that is interpolated in the third term. The bullet list immediately below Eq. (10) describes the intended term as noise padding for r ≥ ⌊k⌋+1, so the displayed equation and its own explanation disagree. Consequently Ψ_k does not equal the discrete flow φ_k of Eq. (9) at integer k, and the stated endpoint property is false. Since Eq. (11) and the loss Eq. (12) are derived from this path, the central derivation is not currently valid.","section":"Section 3.1, Eq. (10)"},{"comment":"Eq. (11) is not the derivative of Eq. (10). Within the active band, differentiating the third term of Eq. (10) with respect to k gives μ′(t)(Fφ − ε), not μ′(t)(ε − Fφ); the printed expression has the opposite sign. In addition, if the noise-padding term were corrected to I_{r≥⌊k⌋+1}ε as the surrounding text suggests, its derivative with respect to k is not included, and the localized vector field shown in Eq. (11) is not a complete derivative of any version of Eq. (10) printed in the paper.","section":"Section 3.1, Eq. (11)"},{"comment":"The stated conditions μ(0)=μ(1)=1 make the active band equal to Fφ at both endpoints of every interval, so at integer k the active band contains data rather than noise. A noise-to-data interpolant over the active band requires μ(0)=0 and μ(1)=1, with endpoint derivative compatibility conditions chosen so that the interpolant matches the discrete flow at integer k and is differentiable in k. The printed condition μ′(0)=−μ′(1) does not provide this, and the sentence below Eq. (10) claiming that the antisymmetric derivative makes Ψ_k differentiable is therefore not justified.","section":"Section 3.1, bump-function conditions"},{"comment":"The training objective in Eq. (12) is not the conditional flow-matching objective introduced in Eq. (8). It is written as an expectation over φ0 of an integral with an unexplained factor dφ0 dk, and it does not explicitly average over the noise ε that conditions the vector field in Eq. (11). The correct conditional flow-matching loss should be an expectation over joint pairs (φ, ε), or a factorization with a conditional noise distribution, followed by an integral over k of ‖v_k(Ψ_k;θ) − dΨ_k/dk‖². As printed, the conditioning structure that defines K-Flow disappears from the loss.","section":"Section 3.1, Eq. (12)"},{"comment":"The factorization in Eq. (15) is asserted without derivation and is not a consequence of Eq. (10) as printed. For a general invertible transform F, conditional independence of the high-band noise from the low-band data is not automatic and must follow from the exact structure of the interpolant; with the noise term in Eq. (10) double-counting low and active bands, the claimed product form is not verifiable. This matters because the steerability experiments in Section 5.4 are motivated by this factorization, so the theoretical support for those experiments is currently missing.","section":"Section 4.1, Eq. (15)"}],"minor_comments":[{"comment":"The sentence 'Table 2 summarizes the results on LSUN Church' appears to refer to Table 3, since the LSUN Church results are reported in Table 3.","section":"Section 5.1, Tables 2 and 3"},{"comment":"In Eq. (8), the integration bound is the same letter t as the dummy time variable, and the integral should run over the full interval [0,1]; as written the expression is not well formed.","section":"Section 2.3, Eq. (8)"},{"comment":"The main derivation uses k∈[0,K], but Algorithm 1 normalizes k←k/kmax and samples k∼U(0,1); the normalization should be stated in the derivation because it rescales the vector field and affects the strength of the loss.","section":"Section 3.1 and Algorithm 1"},{"comment":"The bump function μ(t) is never specified in the experiments; the authors should report the explicit form used to produce Tables 2-5, since the paper emphasizes the role of μ in differentiability and localization.","section":"Section 3.3 and Algorithm 1"},{"comment":"Figure 3 does not label its axes and does not state the normalization of the 'relative norm distribution', which makes the low-scaling hypothesis difficult to verify from the figure.","section":"Figure 3"},{"comment":"The heading 'Date-dependent PCA Transform' should read 'Data-dependent PCA Transform'.","section":"Section 3.2"}],"recommendation":"major_revision","confidential_remarks":"The main concern is internal consistency rather than disagreement with prior work. I did not find evidence of circular reasoning: the method is a construction plus empirical comparison, and the molecular section builds on AssembleFlow but reports standard metrics against independent baselines. The load-bearing derivation errors in Section 3.1 are repairable in principle, but they affect the central claim and must be corrected before the paper can be reconsidered. The authors should also consider releasing code and experimental configuration details, since the plug-in claim is central to the paper's practical relevance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the idea: replacing flow-matching time with a frequency-scale parameter is a genuinely new combination, and the three K-amplitude decompositions (Fourier, wavelet, PCA) give an architecture-agnostic recipe with a localized vector field. The empirical results are competitive, and the scale-steering edits are the most interesting part. That is real value.\n\nNow the soft spots. The stress-test is correct. Eq. (10) has the noise term (1 − I_{r≥⌊k⌋+1})ε, which is active on all coefficients below ⌊k⌋+1, including the low bands already set to Fφ and the active band. At integer k, with μ(0)=1, the active band is Fφ+ε, not Fφ, so Ψ_k never equals the discrete φ_k. The bullet list right below the equation describes the intended term as I_{r≥⌊k⌋+1}ε, so the equation contradicts its own explanation. Eq. (11) is then not the derivative of (10); it also has the sign of the active-band update flipped and drops the noise contribution. On top of that, μ(0)=μ(1)=1 means each scale band starts and ends at data, so the path is data→noise→data rather than noise→data. These are internal inconsistencies in the load-bearing construction, not disagreements about interpretation. No code is provided, and training details are thin, so I can't tell whether the implementation actually used a corrected interpolant.\n\nTo be clear, this looks repairable: change the second term to I_{r≥⌊k⌋+1}ε, set μ(0)=0, μ(1)=1, and re-derive the velocity. If that is what the experiments ran, the paper has a publishable core. As written, the derivation does not hold.\n\nOne minor point: the conclusion says the method was verified exclusively on images, right after a molecular assembly section; that looks like a leftover sentence. Also the molecular baseline is AssembleFlow, with overlapping authorship, which is fine, but the comparison should be checked carefully.\n\nWho is this for? Someone working on multi-scale generation, wavelet diffusion, or steerable flow models. They would find the scale-as-time perspective useful even if this draft is not the right reference. I would not cite it in its current form. But I would send it to a referee: the error is concrete and central, and the fix is small enough that a serious referee could quickly determine whether the corrected version works. If the math checks after correction, this could become a solid paper.","headline":"Scale-as-time flow matching is a genuinely fresh idea, but Eq. (10) as written never produces the claimed noise-to-data path, so the central derivation needs a corrected version before the paper can be trusted.","tokens_in":21476,"tokens_out":3700,"would_cite":false,"duration_ms":36589,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"K-Flow replaces the flow-matching clock with a scale axis, letting generation be steered by frequency band while matching competitive image and molecule quality.","keywords":["K-Flow","flow matching","K-amplitude decomposition","wavelet transform","Fourier transform","PCA","multi-scale generation","molecular assembly"],"falsifier":"Evaluate Eq. (10) at an integer k, say k = 1, for a one-dimensional two-band signal and compare it with the discrete sample $\\varphi_k$ defined in Eq. (9); for a valid interpolant the two must agree at every integer k, and a finite-difference derivative of Eq. (10) across a fractional interval must equal Eq. (11). Because the second noise term in Eq. (10) is written as $\\left(1 - I_{r\\ge \\lfloor k\\rfloor+1}\\right)\\epsilon$, this equality can fail on the already-filled low bands; correcting that term to $I_{r\\ge \\lfloor k\\rfloor+1}\\epsilon$ and rerunning the check would settle whether the localized vector field and the training loss describe the same path.","tokens_in":1963,"feed_emoji":"🎛️","tokens_out":2086,"duration_ms":118429,"temperature":0.7,"pith_summary":"K-Flow proposes that generative modeling can run along a scale axis rather than an abstract clock: instead of matching a noise-to-data flow in time, it matches a flow in the scaling parameter that indexes frequency bands or principal components of the data. The paper constructs a continuous interpolant that fills a data sample in from low to high bands, leaves untouched bands as noise, and defines a conditional vector field localized to the band being filled. Training with conditional flow matching makes the flow learn each scale's transport from noise to data, and decoding the result gives generated samples. The payoff claimed is explicit, controllable generation: preserving high-scale noise while varying low-scale noise edits coarse semantics, and the reported image and molecule experiments are competitive with standard latent flow matching. A broader claim is that any invertible transform with a one-dimensional scale ordering, such as Fourier, wavelet, or PCA, can serve as the generative time axis.","feed_headline":"K-Flow makes scale itself the flow-matching clock","feed_subtitle":"By indexing generation with frequency bands instead of time, K-Flow lets you steer which image details to keep.","key_machinery":"The central object is the K-amplitude interpolant, Eq. (10): an invertible transform $\\mathcal{F}$ maps the signal into scale-indexed bands, and $\\Psi_k$ combines already-filled low bands of data, noise-padded high bands, and a single active band interpolated by the bump function $\\mu(t)$ with $t = k - \\lfloor k \\rfloor$. The bump's derivative antisymmetry makes the path differentiable in $k$, and for linear transforms its conditional derivative, Eq. (11), is nonzero only inside the active band. This localization is what limits the network's velocity field to one band per step and what lets the frozen noise in inactive bands act as a controllable condition during generation.","core_discovery":"The paper's central claim is that the flow-matching clock can be identified with a scale index. For an invertible transform that groups a signal's coefficients by a one-dimensional scaling parameter $k$, K-Flow defines an interpolant $\\Psi_k$ in the K-amplitude space: coefficients with scale below $\\lfloor k \\rfloor$ are set to the data's coefficients, coefficients beyond the next integer band are seeded with independent noise, and the one active band in between is driven from noise to data by a bump function $\\mu(t)$ with $\\mu(0)=\\mu(1)=1$ and $\\mu'(0)=-\\mu'(1)$. For linear transforms the conditional derivative $d\\Psi_k/dk$ is nonzero only on that active band, so the learned velocity field is localized and learns the transport of one band at a time. Training with the conditional flow-matching loss over uniformly sampled $k$ therefore learns the unconditional transport from pure noise to data, and because inactive bands are frozen during each stage, their noise seeds act as conditions that allow scale-wise steering of generated content.","pith_inferences":["One extension the paper does not pursue is to exploit the localization of Eq. (11) architecturally: because the velocity is nonzero only on the active band, a conditional computation model could evaluate the network on that band alone and keep the remaining coefficients frozen, cutting per-step cost.","The paper's observation that low scales carry most of the energy invites an explicit scheduling rule it stops short of deriving: sample discrete scale steps proportionally to band energy, or choose the bump function per band, so compute follows energy.","Since the paper reports both low-to-high and reversed high-to-low flows, a natural generalization is choosing flow direction per task; high-frequency-first generation would be a testable target in medical or scientific imaging where fine detail is primary.","The reported failure of PCA-based K-Flow to give human-interpretable edits suggests scale ordering, not just invertibility, determines steerability; learning the band ordering itself from data would be a concrete next step that this paper leaves open."],"forward_implications":["Because the construction is independent of the specific invertible transform, any one-dimensional scale ordering of a complete basis, such as Fourier, wavelet, or PCA, yields a valid flow-matching generative model, and the paper demonstrates all three.","The flow path is scale-localized, so at each step only one frequency band contributes to the velocity; this allows computational effort to be concentrated on active bands and more steps to be spent on low scales, where image energy concentrates.","Inactive bands' noise stays fixed during each stage, so generation becomes steerable without fine-tuning: fixing high-scale noise edits low-scale content, and fixing low-scale noise edits high-scale details.","Class-conditional generation inherits the same localization: the reported class-dropping behavior indicates that class information mainly steers early, low-scale stages, so later stages can run with the condition removed.","In molecular assembly, combining K-Flow with spectral decomposition of pairwise distances on top of rigid-frame flow matching improves packing matching relative to the compared crystal flow and diffusion baselines."],"supporting_citations":[{"why":"It supplies the conditional flow-matching objective and the equivalence between conditional and unconditional vector fields that K-Flow trains on.","marker":"[9]"},{"why":"It supplies the stochastic-interpolant view that motivates replacing the time variable with the scaling parameter k in Eq. (10).","marker":"[26]"},{"why":"It supplies the latent flow matching baseline and the DiT backbone whose time embedding K-Flow swaps for a K-amplitude embedding.","marker":"[53]"},{"why":"It supplies the VAE latent space and the latent diffusion baseline used in the image generation comparisons.","marker":"[54]"},{"why":"It provides the wavelet-diffusion baseline that K-Flow positions against in frequency-space generative modeling.","marker":"[42]"},{"why":"Its lemma is used to derive the explicit unconditional vector field in Section 4.1.","marker":"[33]"},{"why":"It provides the next-scale autoregressive baseline that K-Flow compares with and that can be read as a discretization of the K-Flow ODE.","marker":"[23]"},{"why":"It supplies the rigid-frame flow-matching machinery for molecular assembly that K-Flow extends with spectral decomposition of pairwise distances.","marker":"[61]"},{"why":"It supplies the COD-Cluster17 crystallization dataset and the CrystalFlow and CrystalSDE baselines for the molecular assembly experiments.","marker":"[58]"}],"fun_headline_variants":["K-Flow: scale becomes the generative clock","Flow matching on frequency, not time: K-Flow","Steer image detail with K-Flow's scale axis","K-Flow turns scale into a flow-matching timescale","Scale-wise flow: K-Flow's new generation dial"],"cache_read_input_tokens":23424,"weakest_assumption_plain":"The load-bearing premise is that Eq. (10) defines a genuine stochastic interpolant: at each integer k it equals the discrete construction with low bands filled with data and higher bands pure noise, and at fractional k its derivative is Eq. (11); if the second noise term in Eq. (10) also touches bands that already carry data or the active interpolated band, the path and the learned velocity field describe different processes.","fun_headline_variants_meta":{"raw":{"variants":["K-Flow: scale becomes the generative clock","Flow matching on frequency, not time: K-Flow","Steer image detail with K-Flow's scale axis","K-Flow turns scale into a flow-matching timescale","Scale-wise flow: K-Flow's new generation dial"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000206,"raw_usage":{"total_tokens":1380,"prompt_tokens":915,"completion_tokens":465,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":385}},"tokens_in":531,"tokens_out":465,"duration_ms":5552,"temperature":1.0,"reasoning_tokens":385,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:55:47.022303+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate Eq. (10) at an integer k, say k = 1, for a one-dimensional two-band signal and compare it with the discrete sample $\\varphi_k$ defined in Eq. (9); for a valid interpolant the two must agree at every integer k, and a finite-difference derivative of Eq. (10) across a fractional interval must equal Eq. (11). Because the second noise term in Eq. (10) is written as $\\left(1 - I_{r\\ge \\lfloor k\\rfloor+1}\\right)\\epsilon$, this equality can fail on the already-filled low bands; correcting that term to $I_{r\\ge \\lfloor k\\rfloor+1}\\epsilon$ and rerunning the check would settle whether the localized vector field and the training loss describe the same path.","supporting_citations":[{"cited_title":"& Ommer, B","cited_arxiv_id":null,"evidence_quote":"It supplies the VAE latent space and the latent diffusion baseline used in the image generation comparisons."},{"cited_title":"& Tran, A","cited_arxiv_id":null,"evidence_quote":"It provides the wavelet-diffusion baseline that K-Flow positions against in frequency-space generative modeling."},{"cited_title":"& Liu, S","cited_arxiv_id":null,"evidence_quote":"It supplies the rigid-frame flow-matching machinery for molecular assembly that K-Flow extends with spectral decomposition of pairwise distances."},{"cited_title":"& Anandkumar, A","cited_arxiv_id":null,"evidence_quote":"It supplies the COD-Cluster17 crystallization dataset and the CrystalFlow and CrystalSDE baselines for the molecular assembly experiments."}],"review_version":1}