{"id":"a304a4a0-f912-4877-9d5b-1b6f7f977b4b","arxiv_id":"2506.08519","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":8,"one_line_summary":"A dynamic graph is approximated as a linear combination of latent adjacency matrices, estimated jointly from partial topology and smooth node signals, with better missing-edge reconstruction than tensor baselines.","lead":"The authors model a time-varying network as a weighted sum of a small number of latent adjacency matrices, modulated by temporal coefficients, and fit these factors from partial edge observations plus node signals assumed smooth on the graph. Their experiments indicate the method reconstructs unobserved edges better than standard tensor decompositions when few connections are observed.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1's convergence guarantee is unproven for the implemented algorithm: the inner ADMM loops take a single projected-gradient step per iteration and Appendix B never verifies the block-coordinate-descent assumptions, so the abstract's central 'prove convergence' claim collapses.","rationale":"The reader's verdict is REJECT, and I agree. I focus on the convergence guarantee rather than the signal-smoothness evaluation because the abstract explicitly advertises a proof of convergence to a stationary point; if that proof is unsound, the paper's theoretical contribution fails even in the hypothetical case where the experiments are clean. The self-confirming ground-truth construction for SeaSurf and USTemp in Section V.A is also a serious validity threat: the target graphs are selected by the same smoothness criterion used in the signal prior (6), so the reported advantage over NSDGD and SGL may be inflated. But the convergence gap is more decisive because Appendix B does not even engage with the relevant assumptions of [46]; it asserts block multiconvexity and continuity and then concludes convergence. The inner ADMM updates are not exact block minimizers, and no convergence analysis of the inexact updates is provided. This is a correctness risk in the method's foundation, not merely a limitation. A concrete computational check can settle whether the implemented inner loop is equivalent to the block-coordinate-minimizer assumed in the theorem.","tokens_in":19347,"tokens_out":10322,"duration_ms":129547,"concrete_test":"Fix one outer iteration on the SwDyn dataset and run the inner ADMM for K=1, 10, 100, and 1000; for each K, compare the A_r update with the exact solution of the convex subproblem (14) obtained from a QP solver. If the projected-gradient ADMM does not converge to the exact minimizer, Proposition 1 cannot hold for Algorithm 1. Independently, check whether the single projected-gradient update (17) satisfies the sufficient-decrease or global-minimization condition required by Theorem 2.3 of [46]; if it does not, the proof in Appendix B is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 1 claims Algorithm 1 converges to a block-coordinate minimizer of (11). Appendix B invokes Theorem 2.3 and Assumptions 1 and 2 of Xu & Yin [46], but verifies only block multiconvexity, convex feasible sets, finite minima, and continuity of set maps. It never checks the assumption that each block update is an exact (or sufficiently accurate) minimizer of the corresponding subproblem. The implemented updates (17) and (26) are single projected-gradient steps on the augmented Lagrangian, not minimizations of (14) or (23). With finite K, as used in all experiments, the inner ADMM does not produce the exact block minimizer; with K tending to infinity, the paper provides no proof that this inexact projected-gradient ADMM converges to the solution of the convex subproblem. Therefore the sequence analyzed in Proposition 1 is not the sequence generated by Algorithm 1, and the central claim 'prove that our approach converges to a stationary point' is unsupported. The issue is not cosmetic: Assumption 1 only ensures strong convexity of the quadratic; it says nothing about the inner iterates of the ADMM scheme.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Dynamic Graph Decomposition (DGD), a model in which the time-varying adjacency tensor of a dynamic network is represented as a sum of R latent graph adjacency matrices A_r modulated by temporal signature vectors c_r. The formulation combines a masked least-squares fit of the observed topology with priors for graph sparsity, signal smoothness, non-overlapping latent supports, temporal smoothness, and a minimum-degree constraint. The authors solve the resulting nonconvex problem by alternating minimization, using inner projected-gradient/ADMM loops for the A_r and C blocks, and claim in Proposition 1 that the algorithm converges to a block-coordinate minimizer (stationary point) of the problem. They also present experiments on synthetic and three real datasets, reporting that DGD outperforms tensor-decomposition baselines and signal-only baselines in reconstructing unobserved edges, especially under limited observations.","tokens_in":19619,"tokens_out":17167,"duration_ms":196659,"significance":"If the formulation and convergence claims were correct, the paper would contribute a useful bridge between tensor decomposition and graph signal processing for dynamic network completion: the latent graphs are interpretable, and the signal-smoothness prior offers a principled way to use nodal observations when topology is partially missing. The synthetic experiments with known latent graphs are a good idea, and the comparison against CPD, BTD, SGL, and a no-signal ablation is informative. However, the paper's central technical claims are not currently supported: the convergence proof does not cover the implemented algorithm, and the gradients used in the algorithm do not match the stated Hadamard-masked objective. These issues are load-bearing, because they affect both the theoretical guarantee and the validity of the empirical evaluation. The core decomposition idea is promising and potentially salvageable, which is why I am not recommending rejection outright, but the manuscript requires substantial revision.","major_comments":[{"comment":"The proof of Proposition 1 does not apply to Algorithm 1 as implemented. Algorithm 1 updates A_r and C via single projected-gradient steps, Eqs. (17) and (26), inside finite-K inner loops; it does not compute an exact minimizer of the block subproblems (14) and (23). Appendix B cites Theorem 2.3 and Assumptions 1–2 of Xu and Yin [46] and verifies block multiconvexity, convexity of feasible sets, finiteness of minima, and continuity of set maps, but it never verifies the key assumption that each block update is an exact (or sufficiently accurate) minimizer. With finite K, the generated sequence is not a block-coordinate minimization sequence; with K tending to infinity, the paper provides no proof that the projected-gradient ADMM loop converges to the solution of the convex subproblem. Moreover, Proposition 1's phrase 'as K→∞' does not address the outer iterations I→∞. The result therefore does not establish convergence of the sequence actually produced by Algorithm 1 to a stationary point of Problem (11).","section":"§IV-A and Appendix B (Proposition 1)"},{"comment":"The updates in Algorithm 1 do not minimize the objective stated in Eq. (11). For the C-subproblem, the loss in Eq. (22) is the Hadamard-masked term 1/2||M0∘(A_vec − A0 C^T)||_F^2, whose gradient with respect to C is −A0^T(M0∘(A_vec − A0 C^T)). Equation (39) instead uses F(−A_vec^T A0 + C A0^T A0) with F=diag(1^T_{N^2} M0), which is the gradient of a row-weighted full-observation loss 1/2||(A_vec − A0 C^T)F^{1/2}||_F^2 and agrees with the true gradient only when the mask is constant across entries. The same problem appears in Eq. (13), where the per-entry mask is summarized by the scalars 1^T_{N^2}m_t rather than applied entry-wise; the quadratic and cross terms in A_r are therefore not those of Eq. (11). As written, the implemented Algorithm 1 solves a different objective, so the convergence proof, even if repaired, would not apply to the reported algorithm.","section":"§III-B, §III-C, and Appendix C (gradients)"},{"comment":"The real-data evaluation is partly self-confirming for the signal-smoothness prior. For SeaSurf and USTemp, the ground-truth graphs are constructed by selecting the top edges where the associated signal is smooth, using odd time measurements; DGD's prior g(X,A,C) in Eq. (6) enforces exactly this smoothness property on the estimated topology, and the method is then evaluated on even time measurements. The target is thus generated under the same smoothness model that the method optimizes. This does not make the comparison logically circular, since the held-out even measurements are separate, but it means the experiments cannot test the validity of the smoothness assumption, and it systematically favors DGD over methods that do not use signal smoothness. The claim that DGD recovers the 'true' latent graphs on these datasets should therefore be qualified.","section":"§V-A (SeaSurf and USTemp ground truth)"}],"minor_comments":[{"comment":"The temporal difference matrix D is described as T×(T−1), but it is used as a (T−1)×T operator in ||DC||_F^2 and in D^T D; please correct the dimensions and notation.","section":"Eqs. (10) and (37)"},{"comment":"Equation (26) writes C^k = P_{S_A}(...), but the projection should be onto the nonnegative orthant S_C, not onto S_A.","section":"Eq. (26)"},{"comment":"The abstract states that DGD outperforms 'signal-based topology identification methods,' but Figure 8 reports that SGL outperforms DGD on the SwDyn dataset for 10–40% observed topology. The abstract should be qualified to reflect this exception.","section":"Abstract and §V-C"},{"comment":"The inner loops use fixed iteration count K and outer loop count I, but the paper does not specify a stopping criterion or convergence tolerance; please state how K and I are chosen in the experiments and how the results depend on them.","section":"Algorithm 1"},{"comment":"The paper refers to a code repository but does not provide a URL or data-availability statement; since the experimental protocol depends on the exact implementation of the mask and the ADMM loops, a public code link would improve reproducibility.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The reader's report recommends rejection, and I agree that the convergence proof and the gradient derivations are serious technical defects. I nevertheless recommend major revision rather than reject because the underlying decomposition idea is coherent and the empirical comparisons are potentially valuable if the algorithm is corrected. The authors need to either (i) prove convergence for the actual projected-gradient ADMM updates, or (ii) change the algorithm to exact block minimization and reprove the result; and they must correct the mask handling in the gradients or explicitly redefine the objective to match the implemented updates. The real-data evaluation also needs an independent ground-truth construction or a clear discussion of the self-confirming nature of the smoothness-based target graphs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Take a look at this if you work on dynamic graph learning. The core idea is a two-way tensor decomposition A = sum_r A_r ⋄ c_r into latent adjacency matrices and temporal signatures, which the authors introduced at ICASSP 2024. What's new here is the signal smoothness coupling (Eq. 6), the non-overlap and degree constraints, and the application to reconstructing unobserved topology from partial edges plus smooth node signals. That is a reasonable and useful extension.\n\nWhat the paper does well: the experiments are fairly solid. DGD beats CPD, BTD, SICA, and the no-signal variant NSDGD across datasets, and the gap grows when observations are scarce. The Contact dataset is an honest independent check—ground truth comes from real interactions, not from the same smoothness prior—and DGD does well there. The authors also report when SGL beats them on SwDyn and give a sensible explanation. That is good scientific behavior.\n\nThe soft spots are real, though. Proposition 1 claims convergence to a stationary point, but Appendix B only checks multiconvexity, feasibility, and continuity; it never verifies that the implemented ADMM updates—single projected-gradient steps on the augmented Lagrangian—actually solve the block subproblems, which is what the cited Xu-Yin theorem requires. With finite K (as used everywhere), the inner loop does not produce exact minimizers; with K→∞, there is no proof that this inexact scheme converges to the subproblem solution. So the analyzed sequence is not the one the algorithm runs. This is a load-bearing gap in the abstract's central claim, and it should be fixed or the claim downgraded to empirical convergence.\n\nThe evaluation on SeaSurf and USTemp also builds the ground-truth graphs by selecting edges where the signal is smooth—the same criterion the method uses as a prior. The temporal train/test split is a plus, but the ground truth itself is not independent, which cheats the comparison in the method's favor. I'd like to see the authors either add another independent ground-truth dataset or explicitly show results on Contact as the primary evidence. The smoothness assumption itself is not stress-tested; if the signals aren't smooth, the prior can pull edges the wrong way.\n\nWho should read this? Researchers in graph signal processing and dynamic network inference. The method is plausible and the empirical comparisons are useful. I would send it to review with the expectation of major revision, mainly to fix or weaken the convergence claim and to de-confound the real-data evaluation.","headline":"A useful signal-aware extension of the authors' ICASSP latent-graph decomposition, but the convergence proof as written does not cover the implemented ADMM steps and the real-data evaluation is partly self-confirming; worth reviewing with major revision.","tokens_in":20180,"tokens_out":3858,"would_cite":true,"duration_ms":43902,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C82","15A69","90C26","94A12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes modeling a dynamic network's adjacency tensor as a weighted sum of a few latent graphs, and claims smooth node signals make that decomposition recover unobserved edges with a provably convergent alternating estimator.","keywords":["dynamic graph learning","latent graph decomposition","temporal networks","signal smoothness","tensor decomposition","alternating minimization","partial observations","network reconstruction"],"falsifier":"Run DGD and its no-signal variant NSDGD on a synthetic dynamic network whose node signals are deliberately generated to be non-smooth with respect to the true graph (for example, white noise projected onto the high-pass Laplacian eigenvectors), keeping 20–40 percent of the edges observed; if DGD does not beat NSDGD in masked-edge relative error, the reported gains come from the smoothness prior rather than from the decomposition structure itself.","tokens_in":19135,"feed_emoji":"🔗","tokens_out":12265,"duration_ms":136170,"temperature":0.7,"pith_summary":"This paper claims that the evolution of a dynamic network can be represented as a small set of latent graphs whose weights change over time, and that node signals observed alongside the topology help recover those components when most edges are unobserved. The proposed Dynamic Graph Decomposition (DGD) writes the time-varying adjacency tensor as a sum of outer products of latent adjacency matrices and temporal signature vectors, then estimates both by alternating convex updates. If the claim is right, partially observed social, sensor, and climate networks can be reconstructed by exploiting the coupling between topology and signal smoothness, and the recovered components are directly interpretable as graphs. The numerical study reports that DGD outperforms standard tensor decompositions and signal-only topology inference, with the largest gains when only 20–40 percent of the topology is observed.","feed_headline":"Latent graphs plus smooth signals recover missing network edges","feed_subtitle":"Partial topology plus signals on nodes lets latent-graph decomposition beat tensor and signal-only baselines.","key_machinery":"At the core is the outer-product decomposition $\\boldsymbol{\\mathcal A}=\\sum_{r=1}^R \\boldsymbol A_r \\diamond \\boldsymbol c_r$ and its time-slice form $\\boldsymbol A_{:,:,t}=\\sum_r [\\boldsymbol c_r]_t \\boldsymbol A_r$, in which the dynamic topology is a weighted sum of $R$ latent graph adjacency matrices, with $\\diamond$ building the tensor whose $t$-th slice is $[\\boldsymbol c_r]_t \\boldsymbol A_r$. The workhorse objective is Problem (11), which balances five terms: the masked least-squares fit, the $\\ell^1$ sparsity of each latent graph, the signal-smoothness cost $\\sum_t\\sum_r [\\mathbf C]_{t,r}\\operatorname{tr}(\\boldsymbol A_r \\mathbf Z_t)/2$ with $\\mathbf Z_t$ encoding pairwise squared signal differences, the orthogonality penalty $\\sum_{r\\ne \\bar r}\\operatorname{tr}(\\boldsymbol A_r^\\top \\boldsymbol A_{\\bar r})$, and the smoothness penalty $\\|\\mathbf D\\mathbf C\\|_F^2$ on the temporal signatures, together with a positivity/degree constraint. Alternating minimization exploits the block multiconvexity of this objective: each $\\boldsymbol A_r$ update and the $\\mathbf C$ update is strongly convex when the other blocks are fixed, and ADMM performs projected gradient steps, giving the convergence guarantee of Proposition 1 via block-coordinate descent theory.","core_discovery":"Dynamic Graph Decomposition (DGD) models the adjacency tensor $\\boldsymbol{\\mathcal A}\\in\\mathbb{R}^{N\\times N\\times T}$ as $\\boldsymbol{\\mathcal A}=\\sum_{r=1}^R \\boldsymbol A_r \\diamond \\boldsymbol c_r$, so the $t$-th time slice is $\\boldsymbol A_{:,:,t}=\\sum_{r=1}^R [\\boldsymbol c_r]_t\\boldsymbol A_r$: each observed topology is a non-negative linear combination of $R$ sparse, undirected latent graphs scaled by their temporal signatures. The estimation problem minimizes a masked least-squares fit to the observed topology plus sparsity on each $\\boldsymbol A_r$, a quadratic-variation penalty that rewards smoothness of the observed node signals on the reconstructed topology, a penalty that keeps the supports of different latent graphs disjoint, a temporal-difference penalty on the signatures $\\boldsymbol c_r$, and a per-node degree lower bound that prevents trivial zero solutions. Algorithm 1 alternates between $R$ convex $\\boldsymbol A_r$-subproblems and one convex $\\mathbf C$-subproblem, each solved with the alternating direction method of multipliers; Proposition 1 proves that the sequence converges to a stationary point (a block-coordinate minimizer) under a mild condition on the observed mask and signatures. Experiments on a synthetic network that switches between stochastic block models and on sea-surface temperature, US temperature, and face-to-face contact datasets report that DGD reconstructs masked edges with lower relative error than CPD, BTD, SICA, smooth-graph learning, and the same method without the signal prior, with the largest advantage at low observation percentages.","pith_inferences":["The paper leaves implicit that the recovered latent graphs are a compressed, interpretable summary of the network's evolution; a natural next step is to feed them into link prediction or anomaly detection, tasks the introduction cites as motivations but does not evaluate.","Because each block subproblem is convex, a streaming extension is plausible: update latent graphs and temporal signatures incrementally as new time slices arrive, which the authors list as future work.","A testable extension is to run DGD on a dataset with independently recorded edge labels while making the node signals deliberately non-smooth; that would isolate the contribution of the signal prior from the contribution of the latent-graph decomposition itself."],"forward_implications":["With 20–40 percent of the topology observed, DGD reconstructs the unobserved edges with lower relative error than the compared tensor-decomposition, signal-only, and no-signal baselines on the tested datasets.","Every latent graph is useful: no single component reconstructs the network as well as their combination, because the temporal signatures of different components peak during different time windows.","The signal prior contributes most when observations are scarce, since the gap between DGD and its no-signal counterpart narrows as the observed fraction of the topology grows.","The alternating estimator can be run to a stationary point, and in the experiments the objective converges within about twenty iterations for most datasets."],"supporting_citations":[{"why":"It supplies the time-varying graph smoothness prior that becomes the quadratic-variation term coupling signals to latent graphs.","marker":"[6]"},{"why":"It defines canonical polyadic decomposition, the standard tensor baseline DGD is compared against.","marker":"[24]"},{"why":"It defines block-term tensor decomposition, the second tensor baseline in the reconstruction experiments.","marker":"[26]"},{"why":"It establishes graph topology identification from smooth graph signals, motivating the signal prior in the objective.","marker":"[37]"},{"why":"It provides the alternating direction method of multipliers used for the convex inner updates.","marker":"[44]"},{"why":"It supplies the block-coordinate descent convergence conditions that Proposition 1 verifies for the alternating scheme.","marker":"[46]"},{"why":"It provides the smooth-graph-learning baseline SGL and the Laplacian low-pass filtering used to generate synthetic smooth signals.","marker":"[47]"},{"why":"It is the preliminary conference version of this work, extended here with the signal prior, convergence analysis, and real-data evaluation.","marker":"[1]"}],"fun_headline_variants":["Signal-smooth latent graphs answer missing network edges","Latent graphs with signal smoothness recover missing edges","Decompose dynamic nets via signal-aware latent graphs","Node signals coupled to latent graphs fix missing topology","Sparse observations? Signal-aware latent graphs reconstruct edges"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the node signals are smooth on the true dynamic graph at every time step, so that connected nodes carry similar values; the paper relies on this assumption both to steer the latent graphs toward correct edges and, for the SeaSurf and USTemp datasets, to construct the ground-truth graphs it evaluates against.","fun_headline_variants_meta":{"raw":{"variants":["Signal-smooth latent graphs answer missing network edges","Latent graphs with signal smoothness recover missing edges","Decompose dynamic nets via signal-aware latent graphs","Node signals coupled to latent graphs fix missing topology","Sparse observations? Signal-aware latent graphs reconstruct edges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000461,"raw_usage":{"total_tokens":2400,"prompt_tokens":1129,"completion_tokens":1271,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":745,"completion_tokens_details":{"reasoning_tokens":1197}},"tokens_in":745,"tokens_out":1271,"duration_ms":12027,"temperature":1.0,"reasoning_tokens":1197,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:09:02.059717+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run DGD and its no-signal variant NSDGD on a synthetic dynamic network whose node signals are deliberately generated to be non-smooth with respect to the true graph (for example, white noise projected onto the high-pass Laplacian eigenvectors), keeping 20–40 percent of the edges observed; if DGD does not beat NSDGD in masked-edge relative error, the reported gains come from the smoothness prior rather than from the decomposition structure itself.","supporting_citations":[{"cited_title":"Learning time varying graphs,","cited_arxiv_id":null,"evidence_quote":"It supplies the time-varying graph smoothness prior that becomes the quadratic-variation term coupling signals to latent graphs."},{"cited_title":"Tensor decompositions and applications,","cited_arxiv_id":null,"evidence_quote":"It defines canonical polyadic decomposition, the standard tensor baseline DGD is compared against."},{"cited_title":"Decompositions of a higher-order tensor in block terms—part iii: Alternating least squares algorithms,","cited_arxiv_id":null,"evidence_quote":"It defines block-term tensor decomposition, the second tensor baseline in the reconstruction experiments."},{"cited_title":"Connecting the Dots: Identifying Network Structure via Graph Signal Processing,","cited_arxiv_id":null,"evidence_quote":"It establishes graph topology identification from smooth graph signals, motivating the signal prior in the objective."},{"cited_title":"Distributed optimization and statistical learning via the alternating direction method of multipliers,","cited_arxiv_id":null,"evidence_quote":"It provides the alternating direction method of multipliers used for the convex inner updates."},{"cited_title":"A block coordinate descent method for regular- ized multiconvex optimization with applications to nonnegative tensor factorization and completion,","cited_arxiv_id":null,"evidence_quote":"It supplies the block-coordinate descent convergence conditions that Proposition 1 verifies for the alternating scheme."},{"cited_title":"How to learn a graph from smooth signals,","cited_arxiv_id":null,"evidence_quote":"It provides the smooth-graph-learning baseline SGL and the Laplacian low-pass filtering used to generate synthetic smooth signals."},{"cited_title":"Tensor graph decomposition for temporal networks,","cited_arxiv_id":null,"evidence_quote":"It is the preliminary conference version of this work, extended here with the signal prior, convergence analysis, and real-data evaluation."}],"review_version":1}