{"id":"fd63c6a8-80c4-4572-8f06-14257ef7a56c","arxiv_id":"2507.21527","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A differentiable joint time-vertex fractional Fourier transform is proposed, with transform orders and Wiener filter coefficients learned by backpropagation for graph signal denoising.","lead":"This paper makes the joint time-vertex fractional Fourier transform differentiable, so transform orders and filter coefficients can be learned by gradient descent instead of grid search. The authors report improved denoising of time-varying graph signals at lower cost, but the evaluation selects the best of 20 random restarts on the test signal.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The stated denoising superiority rests on an unfair experiment: JFRFT-learn is trained and evaluated on the same signal while GNN baselines use an 8:2 train/test split, and only the best SNR over 20 random initializations is reported.","rationale":"I read the paper in good faith. The mathematical construction is a plausible extension of known hyper-differential GFRFT and DFRFT definitions: Eq. (8) defines the joint transform as a Kronecker product, Eqs. (16)-(17) apply the standard matrix-exponential derivative, and the index-additivity and separability properties in Section 3.1 hold formally. The Jordan-decomposition caveat named by the reader is real but it is acknowledged in the limitations and it affects numerical robustness rather than the validity of the adaptive-learning idea. The load-bearing premise of the paper's headline contribution is empirical: adaptive learned orders and filter coefficients are claimed to improve denoising over grid search and over learned graph models. That premise is not tested honestly in the current manuscript. Algorithm 3 optimizes on the very signal whose SNR is reported, and it stores only the maximum over 20 restarts; the GNN baselines are given a train/test split. This data-leakage asymmetry is sufficient to invalidate the performance comparison as it stands. The reader's verdict of CONDITIONAL is therefore appropriate, though the reader's stated weakest assumption (Jordan decomposition stability) is not the point on which the central claim actually turns. The fix is concrete and cheap: rerun with disjoint train/test temporal segments, report mean and variance over restarts, and compare all baselines under an identical protocol. If the advantage disappears, the paper should be revised to claim only that the transform is learnable, not that it outperforms existing methods.","tokens_in":1068,"tokens_out":1073,"duration_ms":72004,"concrete_test":"Re-run the real-data experiments (SST, PM-2.5, COVID) with an explicit train/test split for all methods: use the first 240 time points to fit alpha, beta, and H, and evaluate SNR on the remaining 60 time points, or equivalently on a randomly held-out temporal segment. Run all 20 random initializations, report mean plus/minus standard deviation of test SNR instead of the maximum, and apply the identical split to the GNN, ARMA, GFRFT-search, and JFRFT-search baselines. If the mean held-out SNR of JFRFT-learn still beats JFRFT-search and the GNN baselines by the margins shown in Tables 3-5, the central claim survives. If the margin collapses or reverses, the reported advantage is an artifact of training on the test signal and best-of-N selection.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that learnable JFRFT 'consistently outperforms' existing methods in denoising time-varying graph signals. The evidence for this claim is Tables 3-5 and Algorithm 3. In Algorithm 3, for each random initialization, the network is trained on the full noisy signal, then the SNR is computed on the output of that same signal; after 20 runs, only the maximum SNR is stored (S_NRmax = max over 20 runs). No held-out temporal segment or vertex subset is used for the proposed method. In contrast, the GNN baselines are explicitly given an 8:2 train/test split, with the test set containing 60 time points, and the best test result is selected across hyperparameter configurations. This is not an equal-footing comparison: the reported JFRFT-learn numbers are training-set SNRs, best-of-20, while the reported GNN numbers are held-out test SNRs. The synthetic experiment in Section 4.2.2 has the same structure: noise is generated in the (0.55,0.45) JFRFT domain and the transform orders, filter, and clean signal all come from the same realization, so reaching MSE around 10^-14 demonstrates self-recovery rather than denoising on unseen noise. The claim of improved denoising performance is therefore unsupported by the experiments as reported. A fair protocol could well change the ranking, especially because Hlearn has N*T trainable coefficients and can overfit a single noisy realization.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a learnable joint time-vertex fractional Fourier transform F_J^{α,β} = F^β ⊗ F^α_G using hyper-differential GFRFT and DFRFT, proves basic algebraic properties (reduction, index additivity, commutativity, reversibility, separability), derives gradients with respect to α and β, and embeds transform order pair and diagonal filter coefficients as trainable parameters in a Wiener-filtering network. Experiments cover a transform-order learning sanity check, synthetic and real-world denoising comparisons on SST, PM-2.5 and COVID data, a REDS image enhancement illustration, and a runtime comparison with grid search. The main claimed contributions are adaptive selection of the order pair and filter coefficients without grid search, and improved denoising of time-varying graph signals.","tokens_in":26620,"tokens_out":5303,"duration_ms":63506,"significance":"If the empirical claims were properly supported, the contribution would be a useful model-driven alternative to grid-search JFRFT: gradient-based tuning of the order pair and Wiener coefficients is a natural and computationally attractive extension of earlier trainable GFRFT work. The differentiability calculation in Eqs. (16)-(17) is correct but elementary, and the property proofs are straightforward. The paper also provides a transparency-strengthening artifact: a public code repository link and an explicit limitation list. However, as reported, the denoising experiments do not support the paper's superiority claim because of the training/evaluation protocol mismatch. The theoretical contribution is sound and incremental; the empirical claim is the load-bearing part that needs rework.","major_comments":[{"comment":"The comparison is not on equal footing. In Algorithm 3, for each of 20 runs the network is trained on the full noisy signal Y and the SNR is computed on the output for that same signal, then the maximum over runs is stored (SNRmax, lines 7-29). In contrast, the GNN baselines are explicitly trained on an 8:2 temporal split with a held-out test set of 60 time points ('the test set contains data from 60 time points that need to be denoised'). Thus the reported JFRFT-learn numbers are best-of-20 training-set SNR values, while the GNN numbers are held-out test SNR values. Since the central claim is that the proposed method 'consistently outperforms' other methods, this protocol cannot support that claim. A fair evaluation should use a held-out temporal or vertex split for the proposed method and report mean and standard deviation over initializations rather than the maximum.","section":"Algorithm 3; Tables 3-5"},{"comment":"The synthetic overlap=0 results, with SNR around 130 dB and MSE near 10^-14 for Hfixed and 10^-9 for Hlearn, are self-recovery checks rather than denoising results: the clean signal is generated by the same transform family at (0.55,0.45), and the 'high-frequency' noise is generated in the same JFRFT domain. This demonstrates that the optimizer can invert the transform when the model class contains the true signal, which is a useful sanity check, but it does not show that the method denoises unseen noise. The paragraph describing Table 2 should be reworded to separate this identifiability check from the denoising claim.","section":"Section 4.2.2; Table 2"},{"comment":"The construction relies on an exact and numerically stable Jordan decomposition of the graph shift operator to form log(F_G) and the matrix exponentials in Eqs. (16)-(17). The paper acknowledges in its limitation list that 'for large-scale graphs or when the shift operator is nearly defective, numerical instability during decomposition may adversely affect the filtering performance.' This is not merely a caveat: it limits the applicability of the core model, and the experiments in Tables 3-5 use small k-NN graphs whose shift operators are well-behaved. The robustness claims should be scoped accordingly, and a small experiment on a defective or nearly defective shift operator would make the limitation concrete.","section":"Eq. (4); Section 5, limitation 1"},{"comment":"The computational-efficiency claim is plausible asymptotically, but the reported wall-clock numbers are not internally consistent. In Section 4.2.2 the text states that the learnable JFRFT training 'takes less than a minute,' while Table 8 reports 990 s for JFRFT-learn at N=10, T=10. The reader cannot tell whether the 990 s includes 20 repeated runs, 10,000 epochs each, or whether the minute is for a single run at a different scale. The runtime column should state the exact protocol used for the timing measurement.","section":"Section 4.2.2; Table 8"}],"minor_comments":[{"comment":"The text states that the original transform order pair is (0.55,0.45) and (1.55,1.45), but Table 1's row header reads (0.45,0.55) and the first block learns (0.4500,0.5500); the notation should be made consistent.","section":"Table 1; Section 4.1"},{"comment":"The separability property as written, F_J^{α,β} = F_G^α F^β, mixes operators acting on different dimensions without specifying the action on a matrix signal; writing the identity with the Kronecker structure or with an explicit matrix action would avoid ambiguity.","section":"Eq. (15)"},{"comment":"Lines 21-30 update α, β and H inside the epoch loop, but line 32 says to 'compute new Y0' and update the same parameters again; the pseudo-code should distinguish forward evaluation from parameter updates to avoid redundant or contradictory steps.","section":"Algorithm 2"},{"comment":"Equation (23) uses G_G for the graph transform, which clashes with the graph notation G = (V,A); using distinct symbols for the graph and the transform matrix would improve readability.","section":"Section 2.1; Section 4.2.1"},{"comment":"The proof of index additivity in Eq. (12) uses the commutativity of the temporal DFRFT matrices F^{β1} and F^{β2}; this commutativity holds for the eigen-decomposition definition and should be stated explicitly.","section":"Section 3.1"}],"recommendation":"major_revision","confidential_remarks":"The theoretical part is sound but incremental, and the paper's main advertised value is the empirical denoising superiority. Since that claim currently rests on an unfair train/test protocol and best-of-20 training SNR, the authors should be asked to re-run the real-data experiments with a held-out split and report mean and variance. I would not reject on the current evidence, but I would not accept without the corrected experiments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the core construction is legitimate: the authors form a joint time-vertex fractional Fourier transform as the Kronecker product of a hyper-differential graph fractional Fourier transform and a discrete fractional Fourier transform, then treat the two orders and the diagonal filter coefficients as trainable parameters. The differentiability derivation (Eqs. 16–17) is elementary but correct, and the transform-learning sanity check in Section 4.1 works as advertised, recovering the original order pair even with multiple layers. They also ship code, which is good practice.\n\nSecond, the denoising experiments do not support the headline claim. Algorithm 3 trains on the full noisy signal, computes SNR on the output of that same signal, and repeats 20 times with random initializations, keeping only the maximum SNR. The GNN baselines, in contrast, get an 8:2 train/test split and report held-out test performance. That is not a comparison on equal footing. The reported JFRFT-learn numbers are training-set SNRs, best-of-20, while the GNN numbers are test-set numbers. Hlearn also has N*T trainable coefficients, so it can overfit a single noisy realization. The synthetic experiment in Section 4.2.2 has the same self-referential structure: the noise is generated in the same transform domain, and the filter and clean signal come from the same realization, so MSE near 10^-14 demonstrates self-recovery, not denoising on unseen noise. The authors' own limitation list mentions Jordan decomposition instability for nearly defective shift operators, but that is a minor issue compared to the evaluation protocol.\n\nThe theory section is a fairly direct extension of prior differentiable GFRFT and DFRFT work, but it is the first to combine them into a trainable JFRFT, so it is a reasonable incremental contribution. The property proofs are simple and mostly rest on index additivity of the factors. The complexity comparison with grid search is plausible, and the runtime table is consistent with that.\n\nWho should read this? Someone working on fractional graph transforms or model-driven denoising for time-varying graph signals. The paper deserves a serious referee because the construction is coherent and the evaluation flaw is fixable. My recommendation: send it to peer review, but with a clear request that the authors re-run the real-data experiments with a held-out temporal or vertex split, report mean and variance over restarts, and compare against GNNs on the same data protocol. If they do that, the results may change, and the paper would be much stronger.","headline":"The trainable JFRFT construction is sound and the differentiability math checks out, but the denoising experiments are not on equal footing: the proposed method trains and evaluates on the same signal and reports best-of-20, while GNN baselines use an 8:2 split, so the claimed superiority is unsupported.","tokens_in":27132,"tokens_out":1937,"would_cite":false,"duration_ms":23325,"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 makes the joint time-vertex fractional Fourier transform learnable: it embeds the transform order pair (alpha, beta) and Wiener filter coefficients as trainable parameters of a neural network, optimizes them by gradient…","keywords":["graph signal denoising","joint time-vertex fractional Fourier transform","hyper-differential GFRFT","learnable transform orders","Wiener filtering","backpropagation"],"falsifier":"Repeat the SST, PM-2.5 or COVID experiment on a graph whose shift operator is deliberately made nearly defective, e.g. by adding a small Jordan block perturbation, and compare the learned $\\alpha$, $\\beta$ and output SNR against a well-conditioned graph of the same size: the paper's own limitation implies the SNR should collapse and the learned orders should become unstable.","tokens_in":26121,"feed_emoji":"📉","tokens_out":2720,"duration_ms":26529,"temperature":0.7,"pith_summary":"The paper tries to establish that the joint time-vertex fractional Fourier transform (JFRFT) can be made trainable end-to-end: its two transform orders — one for the graph dimension, one for the time dimension — and the Wiener filter coefficients can be learned from data by gradient descent instead of chosen by grid search. If true, this makes adaptive spatiotemporal filtering of time-varying graph signals practical for larger graphs, since the expensive grid search over order pairs is replaced by a few dozen backpropagation epochs. The claim is supported by synthetic recovery of known transform orders, by consistently higher output SNR than GFRFT-, JFT-, ARMA-, median-, and GNN-based baselines on SST, PM-2.5 and COVID datasets, and by a reported drop in worst-case runtime from hours to minutes at moderate graph sizes.","feed_headline":"Gradient descent replaces grid search for time-vertex graph denoising","feed_subtitle":"A learnable joint fractional Fourier transform picks its own order pair and filter, cutting runtime by orders of magnitude.","key_machinery":"The key object is the hyper-differential JFRFT operator $F_J^{\\alpha,\\beta} = F^\\beta \\otimes F_G^\\alpha$, built from the hyper-differential GFRFT $F^\\alpha_G = \\exp(-j\\alpha\\pi/2 \\cdot (\\pi(D_G^2 + F_GD_G^2F_G^{-1}) - \\tfrac12 I))$ and the discrete FRFT matrix $F^\\beta$. Its role is to define a two-parameter spectral domain in which the transform orders act as continuous, differentiable controls, so that the gradient of the loss with respect to $\\alpha$ and $\\beta$ can be computed and the orders, together with the Wiener filter coefficients, can be optimized by backpropagation.","core_discovery":"The central claim is that the hyper-differential form of the JFRFT, defined as the Kronecker product $F_J^{\\alpha,\\beta} = F^\\beta \\otimes F_G^\\alpha$, is differentiable in both fractional orders, so $\\alpha$ and $\\beta$ can be learned parameters inside a neural network rather than grid-searched values. The authors prove the required derivatives $\\dot{F}^\\beta$ and $\\dot{F}_G^\\alpha$ as matrix exponentials, embed the order pair plus the diagonal Wiener filter $H$ into a model-driven network, and train all of them by backpropagation with the MSE loss. They further show that the learned orders obey index additivity across layers, and that the adaptive scheme matches or beats the grid-searched JFRFT Wiener filter while cutting complexity from $O(N^3 + N^4T^4)$ to $O(N^3 + N^2T^2)$ per epoch.","pith_inferences":["If the method is right, the same backpropagation scheme should transfer to other tasks where a transform order controls a trade-off between domains, such as time-vertex sampling, reconstruction, or compression, with the order pair learned from a task loss instead of a denoising loss.","The consistent $\\beta \\approx -1$ outcome suggests the temporal fractional order may often be redundant or reducible to a sign flip for real data; one could test whether fixing $\\beta = -1$ and learning only $\\alpha$ retains most of the SNR gain.","A natural stress test is to apply the method to graphs that are deliberately nearly defective, where the Jordan decomposition in Eq. (4) becomes ill-conditioned; the paper's own limitation list predicts the learned orders and filters would degrade there."],"forward_implications":["A pipeline for denoising time-varying graph signals no longer needs a grid search over transform order pairs; the optimal pair is found by gradient descent on the data.","The learned orders can be interpreted as data-adapted spectral coordinates: for real-world datasets the method consistently selects $\\beta \\approx -1$, effectively a time-reversal flip, rather than the default $\\beta = 1$.","The same trainable layer can be stacked, and the learned orders add across layers, so network depth can be traded against individual order magnitudes.","The claimed complexity reduction from $O(N^3+N^4T^4)$ to $O(N^3+N^2T^2)$ makes the method feasible on graphs where the grid-search JFRFT is prohibitively slow."],"supporting_citations":[{"why":"Supplies the hyper-differential definition of the GFRFT and the gradient-based adaptive learning of a single transform order that the paper extends to the joint case.","marker":"[11]"},{"why":"Defines the original fractional-power JFRFT that this paper generalizes to the hyper-differential, learnable form.","marker":"[19]"},{"why":"Establishes Wiener filtering in the JFRFT domain with grid-searched order pairs, the baseline whose computational cost the proposed method targets.","marker":"[23]"},{"why":"Shows that the fractional Fourier transform order is differentiable and can be trained inside a neural network, the template for learning the time-side order.","marker":"[38]"},{"why":"Introduces optimal GFRFT-domain Wiener filtering whose transform-order selection the paper compares against and improves on.","marker":"[22]"}],"fun_headline_variants":["Neural network learns joint time-vertex Fourier orders","Learnable fractional transform orders beat grid search for graph signals","Backprop picks optimal transform orders for time-vertex Wiener filtering","Adaptive JFRFT: gradient-based order selection slashes runtime","Model-driven learning tunes fractional orders and filter jointly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction assumes an exact and numerically stable Jordan decomposition of the graph shift operator, which is required to define the hyper-differential GFRFT in Eq. (4); when the shift operator is nearly defective or the graph is large, that decomposition becomes ill-conditioned and the learned orders and filter output become unreliable.","fun_headline_variants_meta":{"raw":{"variants":["Neural network learns joint time-vertex Fourier orders","Learnable fractional transform orders beat grid search for graph signals","Backprop picks optimal transform orders for time-vertex Wiener filtering","Adaptive JFRFT: gradient-based order selection slashes runtime","Model-driven learning tunes fractional orders and filter jointly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000297,"raw_usage":{"total_tokens":1731,"prompt_tokens":961,"completion_tokens":770,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":689}},"tokens_in":577,"tokens_out":770,"duration_ms":8744,"temperature":1.0,"reasoning_tokens":689,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:39:22.043629+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the SST, PM-2.5 or COVID experiment on a graph whose shift operator is deliberately made nearly defective, e.g. by adding a small Jordan block perturbation, and compare the learned $\\alpha$, $\\beta$ and output SNR against a well-conditioned graph of the same size: the paper's own limitation implies the SNR should collapse and the learned orders should become unstable.","supporting_citations":[{"cited_title":"Alikas ¸ifo˘glu, B","cited_arxiv_id":null,"evidence_quote":"Supplies the hyper-differential definition of the GFRFT and the gradient-based adaptive learning of a single transform order that the paper extends to the joint case."},{"cited_title":"Alikas ¸ifo˘glu, B","cited_arxiv_id":null,"evidence_quote":"Defines the original fractional-power JFRFT that this paper generalizes to the hyper-differential, learnable form."},{"cited_title":"Alikas ¸ifo˘glu, B","cited_arxiv_id":null,"evidence_quote":"Establishes Wiener filtering in the JFRFT domain with grid-searched order pairs, the baseline whose computational cost the proposed method targets."},{"cited_title":"Koc ¸, T","cited_arxiv_id":null,"evidence_quote":"Shows that the fractional Fourier transform order is differentiable and can be trained inside a neural network, the template for learning the time-side order."},{"cited_title":"Ozturk, H","cited_arxiv_id":null,"evidence_quote":"Introduces optimal GFRFT-domain Wiener filtering whose transform-order selection the paper compares against and improves on."}],"review_version":1}