{"id":"e6a400f7-4e41-4dce-81c9-91cbe35f0f33","arxiv_id":"2505.23160","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Topo-LMS brings classical least-mean-squares adaptive filtering to edge signals over simplicial complexes, with stability conditions, closed-form steady-state error, and optimal sampling strategies.","lead":"Topo-LMS adapts the classical least-mean-squares filter to streaming signals on the edges of simplicial complexes, explicitly using triangles and higher-order adjacency to learn from network flow data. The authors derive stability and steady-state error formulas, design near-minimal edge sampling schemes, and give centralized, topology-inference, and distributed variants.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's Theorem 1 and the sampling design rest on Assumption 3 (temporally white regressors), which the generative model (8) itself violates because consecutive regressors share delayed edge signals; the closed-form MSD and rate are therefore heuristic rather than guaranteed.","rationale":"The reader's weakest_assumption is the right one. I independently checked the derivation: Eq. (24) uses E[Q(n)eh(n)] = Q E[eh(n)], which requires Q(n) independent of eh(n); Eq. (26) uses the same for the weighted variance; Eqs. (28)-(31) then replace F by Q^T⊗Q^T. All of these are standard in LMS analysis only under the independence assumption or a small-step-size limit. The paper's model (8)-(9) cannot satisfy the assumption because X(n) and X(n−1) share x(n−1),...,x(n−M); this overlap follows directly from the definition of the regressor and is not a minor technicality. I would still not move to REJECT: the authors explicitly flag Assumption 3 as not strictly verified, the finite-step-size correction is O(μ^2) in the standard theory, and the numerical comparisons in Sec. VI are consistent with the formulas. The correct fix is to re-derive or reinterpret the results under the standard small-μ energy-conservation framework, or to add a numerical test with correlated inputs. The distributed Theorem 2 proof is a separate, mostly sound contraction argument, so the main vulnerability is the centralized closed-form analysis. Since the reader already set CONDITIONAL on this same issue, the verdict is unchanged.","tokens_in":21826,"tokens_out":15785,"duration_ms":165267,"concrete_test":"Simulate model (8) on a small 2-complex as in Sec. VI.A, with x(n) generated as a stationary Gaussian vector AR(1) whose temporal correlation ρ is tunable (ρ=0 makes Assumption 3 true; ρ=0.5, 0.9 violate it), while keeping the stationary covariance Cx(0) and hence CX(p) fixed. Run Topo-LMS for step-sizes satisfying (25), e.g., μ=0.05/λmax(CX), μ=0.2/λmax(CX), and μ=0.8/λmax(CX), and compare the empirical steady-state MSD against (μ/2)Tr(G(p)CX(p)^{-1}) from (32). If the relative error grows with ρ and with μ, exceeding the O(μ^2) level predicted by the approximation, then Assumption 3 is load-bearing and Theorem 1 must be restated as a small-step-size/correlated-input approximation; if it does not, the concern does not land.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the independent-regressor assumption used to derive the closed-form analysis. In Sec. III.B, Assumption 3 states that D(n)X(n) is independent of D(l)X(l) for all l≤n. But the generative model (8) defines X(n) in (9) as [x(n), Lu x(n−1), ..., Lu^M x(n−M), Ld x(n−1), ..., Ld^M x(n−M)]. Consecutive regressors overlap in the delayed signal blocks, so Cov(X(n), X(n−1)) contains terms proportional to Cx(1)=E[x(n)x^T(n−1)], which is generally nonzero under Assumption 2. Thus Assumption 3 is violated by the exact model the analysis is meant to describe. This matters because the mean recursion (24), the variance recursion (26), the substitution F≈Q^T⊗Q^T, and hence Theorem 1's MSD formula (32) and convergence rate (34) all use E[Q(n)eh(n)] = Q E[eh(n)] and E[Q^T(n)ΣQ(n)] ≈ Q^TΣQ; without temporal independence these equalities fail, since eh(n) is a function of past regressors that are correlated with X(n). The paper itself concedes this ('Even if Assumption 3 is not strictly verified in model (8)'), and the same issue reappears as Assumption 4 and in the distributed MSD (56)-(60). Because the optimal sampling problem (35)-(37) is calibrated on (32) and (34), a material violation propagates to the sampling probabilities and the claimed MSE/convergence guarantees. The rest of the argument is internally consistent if the assumption is imposed; the weakness is the gap between the stated assumptions and the model.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Topo-LMS, an adaptive least-mean-squares algorithm for estimating the coefficients of simplicial FIR filters from streaming, partially observed edge-flow signals on two-dimensional simplicial complexes. The authors derive mean and mean-square recursions under stated stochastic assumptions, obtain closed-form expressions for the steady-state MSD and convergence rate (Theorem 1), and use these expressions to formulate an edge-sampling probability design problem with a convex relaxation. They further propose an alternating adaptive method for joint filter estimation and triangle (upper adjacency) topology inference, and a distributed diffusion variant of Topo-LMS with its own stability analysis (Theorem 2). The theoretical results are tested on synthetic data and on German DFN traffic measurements, where the proposed methods are compared favorably with a graph-only Edge-Laplacian LMS baseline. The central theoretical claims are the stability condition 0 < mu < 2/lambda_max(C_X), the MSD formula mu/2 Tr(G C_X^{-1}) + O(mu^2), and the rate approximation 1 - 2 mu lambda_min(C_X), together with the sampling design built on them.","tokens_in":22250,"tokens_out":7720,"duration_ms":85965,"significance":"If the central results were fully rigorous for the model in (8), this would be a useful and timely extension of graph-based adaptive LMS methods to higher-order topological domains: it provides a closed-form stability condition, steady-state MSD and convergence-rate analysis, a principled (if approximate) optimal sampling scheme, a joint topology-inference procedure, and a distributed diffusion algorithm. The paper is also clearly written and builds on a sensible modeling framework. It deserves credit for stating explicit assumptions and for including both synthetic and real-data validation of the algorithmic variants. However, the load-bearing theoretical analysis rests on Assumption 3 (temporally white regressors), which the paper itself concedes is not verified by the generative model (8), and the proof of Theorem 1 additionally replaces the exact matrix F by an approximation before computing the infinite sum that defines the steady-state MSD. The sampling design in Section III.C is calibrated entirely on these approximate expressions.","major_comments":[{"comment":"Assumption 3 (temporally white regressors) is violated by the generative model in (8). Since X(n) in (9) is [x(n), Lu x(n-1), ..., Ld^M x(n-M)], consecutive regressors share delayed signal blocks: for example, x(n-1) appears both as the second block of X(n) and as the first block of X(n-1), so Cov(X(n),X(n-1)) contains terms proportional to C_x(1), which is generally nonzero under Assumption 2. The paper explicitly states that 'Even if Assumption 3 is not strictly verified in model (8)'. Because the mean recursion (24), the variance recursion (26), and Theorem 1's MSD formula (32) and rate (34) all rely on the independence of Q(n) from e_h(n) and on replacing E[Q^T(n)Sigma Q(n)] by Q^T Sigma Q, these closed-form results are not guaranteed for the actual data model. The same gap propagates to the sampling constraints (35b)-(35c). I ask the authors to either prove the analysis under assumptions compatible with (8), or explicitly label Theorem 1 and the sampling design as approximate and support this by numerical experiments that quantify the mismatch for correlated regressors.","section":"Sec. III.B, Assumption 3, Eqs. (8)-(9)"},{"comment":"Even granting Assumption 3, the derivation of the closed-form MSD in Theorem 1 replaces F = E[Q^T(n) ⊗ Q^T(n)] by Q^T ⊗ Q^T before computing (I - F)^{-1}. This substitution is stated to be accurate for sufficiently small mu, but no error bound is provided. Consequently, the O(mu^2) remainder in (32) and the rate approximation (34) are formal rather than proven: the exact F depends on fourth-order moments of the flow signals, and the analysis never quantifies the difference between the exact and approximate F. A perturbation bound in terms of mu, the fourth-order cumulants, and the spectral structure of C_X would be needed to turn (29)-(34) into a rigorous theorem. Without such a bound, the 'closed-form' steady-state MSD is an approximation whose accuracy is only asserted.","section":"Sec. III.B and Appendix A, Eqs. (29)-(32), (62)-(64)"},{"comment":"The optimal sampling design is calibrated on the MSD and rate expressions in (32) and (34), whose status is heuristic under the actual model (8) for the reasons given above. Thus constraints (b) and (c) in (35) and (37) do not supply certified guarantees on convergence rate or steady-state MSD for the data model used in the experiments. The numerical validation in Section VI.B also generates data from model (8) and checks only that the achieved MSD satisfies the target gamma; it does not compare the optimized sampling probabilities against alternative sampling schemes, nor does it report the true value of lambda_min(C_X(p)) or the approximation error in (34). I recommend adding such comparisons and explicitly stating which parts of the sampling methodology are heuristic if the theoretical gap remains.","section":"Sec. III.C, Eqs. (35)-(37) and Sec. VI.B"}],"minor_comments":[{"comment":"The notation for matrix rows and columns is garbled: the manuscript prints both the i-th row and the i-th column of A as [A]_i. Please disambiguate, e.g., with [A]_i,: and [A]_:,i.","section":"Notation paragraph"},{"comment":"In the proof of Lemma 1, comparing F(0) <= F(v_i) leads to v_i <= sqrt(2 lambda_0); the text states 'v_i <= sqrt(2 lambda_1)', which is a typo.","section":"Appendix B"},{"comment":"Section VI.A refers to a construction procedure 'from Sec. VA'; the intended cross-reference is to Section VI.A itself. Also, in Section VI.D, the word 'unsing' should be 'using'.","section":"Sec. VI.A and Sec. VI.D"},{"comment":"The ratio Tr(G(p))/lambda_min(C_X(p)) is described as defining a pseudo-convex map whose sublevel sets are convex. The arguments given only establish convexity of the sublevel sets (i.e., quasi-convexity), and the reformulated constraint is directly convex; the terminology should be corrected or the standard convexity argument should be used.","section":"Sec. III.C, Eq. (37)"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically competent and the topic is fitting for a signal processing journal. My main concern is that the paper presents approximate results as theorems: the admitted violation of Assumption 3 and the uncontrolled F-approximation in Appendix A are load-bearing for Theorem 1 and for the sampling problem in Section III.C. I would support acceptance after the authors reframe the central claims as approximate, add quantified numerical evidence for the approximation error, and temper the guarantee language in the abstract and conclusions. No concerns about novelty disclosure beyond the relation to the authors' prior conference paper [1] and the graph-LMS baseline [31]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a competent, incremental extension of graph-LMS to simplicial complexes, with the expected machinery and one admitted caveat. The headline result is a closed-form MSD and convergence-rate analysis for Topo-LMS under the standard independent-regressor assumption, plus sampling, topology-inference, and diffusion variants built on top. The theory is classic LMS analysis; what is new is the simplicial packaging and the sampling-rate optimization.\n\nWhat is genuinely new: Theorem 1 gives a clean closed-form MSD for edge-flow regressors built from Hodge Laplacians. The sampling-probability design under rate constraints is a useful practical tool. The adaptive topology inference via hard-thresholded alternating gradient descent is a reasonable heuristic. The distributed diffusion version naturally extends the centralized method, and Theorem 2's observability condition (one agent with non-singular local covariance) is a nice touch. Synthetic experiments match the steady-state MSD, and the DFN traffic data show modest gains over the graph-only edge-Laplacian baseline.\n\nThe main soft spot is Assumption 3: it assumes D(n)X(n) is temporally white, but the generative model (8) makes consecutive regressors share delayed signals, so the assumption is violated by the model itself. The paper admits this in a parenthetical, yet the closed-form MSD and rate in Theorem 1 are derived from it. That is not fatal, because the independent-regressor assumption is standard in adaptive filtering and small-step-size arguments can justify the formulas as approximations, but the paper should say that explicitly instead of presenting (32) as exact. The optimal sampling problem inherits this approximation. Also minor: the real-data comparison is on a single small network (17 nodes), so the empirical claim is suggestive rather than sweeping.\n\nWho it is for: researchers in topological signal processing and adaptive filtering on graphs will want this. It does not resolve a broad open problem, but it is a solid toolbox contribution with useful theory and experiments.\n\nRecommendation: send it to peer review. It deserves a serious referee. Ask the authors to clarify the status of Assumption 3, state that the MSD formula is a small-step-size approximation with a supporting argument, and consider releasing code to improve reproducibility.","headline":"Solid incremental extension of graph-LMS to simplicial complexes; the analysis is standard, the caveat is admitted, and the paper deserves a serious referee.","tokens_in":22758,"tokens_out":2723,"would_cite":true,"duration_ms":29415,"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":"The paper claims that classical LMS adaptive filters can be lifted from graphs to simplicial complexes, giving an online algorithm to learn and predict streaming edge-flow signals from partially sampled edges.","keywords":["adaptive filtering","least mean squares","simplicial complexes","topological signal processing","Hodge Laplacian","edge flow prediction","sampling strategies","distributed diffusion"],"falsifier":"Run Topo-LMS on data generated by the paper's model (8) with strongly autocorrelated input, e.g. an AR(1) process with coefficient close to one, and compare measured steady-state MSD and convergence rate with formulas (32) and (34); a gap that grows with the autocorrelation would show those closed-form expressions depend essentially on temporal whiteness.","tokens_in":21617,"feed_emoji":"🔺","tokens_out":12229,"duration_ms":112454,"temperature":0.7,"pith_summary":"The paper sets out to show that the classical least-mean-squares (LMS) adaptive filter, which learns a linear model online from streaming data, can be moved from graphs to simplicial complexes: signals living on the edges of a network (traffic, hydraulic, or communication flows) are modeled as outputs of a filter built from the upper and lower Hodge Laplacians, and the filter coefficients are updated sample by sample from partially observed edges. The main theoretical claim is a complete stochastic description of the resulting Topo-LMS recursion: a step-size stability condition, a closed-form steady-state mean-square deviation, and a convergence-rate formula, all expressed through the covariance of the sampled regressors. The paper also claims that edge-sampling probabilities can be optimized by a convex program that minimizes sampling rate under constraints on error and speed, that the latent triangle structure of a complex can be inferred jointly with the filter, and that a distributed diffusion version converges in the mean under a mild network condition. If these claims are right, online flow prediction over higher-order network structure becomes as tractable as graph LMS, and the reported experiments on traffic measurements indicate that using triangle information helps prediction compared with a graph-only LMS baseline.","feed_headline":"Topological LMS tracks streaming flows from partial edge samples","feed_subtitle":"Closed-form stability, steady-state error, and sampling rules make it usable for online traffic-flow prediction.","key_machinery":"The machinery is the pair of Hodge shift operators attached to a two-dimensional simplicial complex (a combinatorial space made of vertices, edges, and triangles): the lower Laplacian $L_d$ propagates an edge signal to edges sharing a vertex, while the upper Laplacian $L_u$ propagates it to edges that co-bound a common triangle. These operators build the regressor matrix $X(n)=[x(n), L_u x(n-1), \\ldots, L_u^M x(n-M), L_d x(n-1), \\ldots, L_d^M x(n-M)]$, whose rows each edge can compute locally. The update $h(n+1)=h(n)+\\mu X^T(n)D(n)(y(n)-X(n)h(n))$ is the classical LMS gradient step with the sampled regressors $D(n)X(n)$. The stochastic analysis then reduces the algorithm to the covariance $C_X(p)=\\mathbb{E}\\{X^T(n)D(n)X(n)\\}$: stability depends on $\\lambda_{\\max}(C_X)$, steady-state error on $\\mathrm{Tr}(G(p)C_X^{-1})$, and convergence speed on $\\lambda_{\\min}(C_X)$, with the Kronecker-product approximation $F\\approx Q^T\\otimes Q^T$, $Q=I-\\mu C_X$, converting the mean-square recursion into the closed-form steady-state expression of Theorem 1.","core_discovery":"On the paper's own terms, the central discovery is that the stochastic recursion $h(n+1)=h(n)+\\mu X^T(n)D(n)(y(n)-X(n)h(n))$ is a valid adaptive estimator for edge-flow signals over a two-dimensional simplicial complex. Under Assumptions 1-3 (independent sampling, wide-sense stationarity, and temporally white regressors $D(n)X(n)$), Theorem 1 states that the recursion converges in the mean whenever $0<\\mu<2/\\lambda_{\\max}(C_X(p))$, that for small $\\mu$ its mean-square convergence rate is $\\alpha\\simeq 1-2\\mu\\lambda_{\\min}(C_X(p))$, and that its steady-state MSD is $\\frac{\\mu}{2}\\mathrm{Tr}(G(p)C_X(p)^{-1})+O(\\mu^2)$. The matrices $C_X(p)$ and $G(p)$ are second-order statistics of the sampled, shifted edge signals, so the formulas connect convergence behavior directly to topology and sampling. The paper further claims that the sampling probabilities $p$ can be chosen by minimizing $\\mathbf{1}^T p$ under MSD and convergence constraints (via a convex relaxation), that an alternating stochastic gradient procedure with a hard-thresholding proximal step recovers true triangle indicators $t^o$, and that the distributed adapt-then-combine version converges in the mean if the communication matrix is irreducible and at least one edge agent has non-singular local covariance (Theorem 2).","pith_inferences":["Editorial extension: the same recursion should carry over to signals on triangles or vertices by replacing $L_u,L_d$ with the Hodge Laplacians of the corresponding order, and the paper states the theory extends to regular cell complexes; testing on triangle-valued flow data would check that generality.","Editorial extension: because the paper concedes that Assumption 3 is not strictly verified by its own lagged model, the closed-form MSD and rate formulas should be treated as small-step-size approximations for strongly correlated inputs; a data-driven check with an AR(1) input would quantify the deviation.","Editorial extension: the sampling optimization is solved offline, but the same objective could be run online to adapt $p$ as signal statistics drift, turning the static design into a control policy for data acquisition.","Editorial extension: the reported gains over graph-only LMS could be probed further by comparing against simplicial vector-autoregressive and Kalman baselines, which would isolate whether triangles or the LMS tracking mechanism drive the improvement."],"forward_implications":["If the central results hold, a practitioner can run Topo-LMS on a partially observed network and know in advance which step sizes are stable and how much steady-state error to expect, using only the sampled covariance $C_X(p)$.","The sampling design becomes an engineering tool: solving the convex relaxation of the sampling problem gives the cheapest set of edges to observe while meeting prescribed targets on convergence speed and steady-state MSD.","When only the graph of the network is known, the alternating algorithm can recover which triples of nodes actually form triangles, and can re-learn that set when the topology changes over time.","The distributed diffusion variant allows prediction to be performed at the edge level with neighbors exchanging only local estimates, and converges in the mean even when most agents have singular local covariance, as long as one agent per connected component observes enough information.","On the evaluated traffic data, the topology-aware filter predicts future edge flows with lower normalized error than the Edge-Laplacian/graph-only LMS baseline, in both centralized and distributed forms."],"supporting_citations":[{"why":"Supplies the Hodge Laplacian definitions, simplicial Fourier transform, and topology-inference tools that define $L_u$ and $L_d$ in the observation model.","marker":"[9]"},{"why":"Supplies the LMS stochastic-analysis machinery, including the independent-regressor assumption and weighted variance recursions used in Section III.","marker":"[23]"},{"why":"Supplies the distributed adaptation and network-optimization framework used for the stability analysis of the distributed variant.","marker":"[24]"},{"why":"Supplies the graph-domain adaptive LMS with optimal sampling strategies that this paper generalizes to simplicial complexes.","marker":"[29]"},{"why":"Supplies the graph-based LMS model and distributed diffusion adaptation that Topo-LMS extends and compares against in the experiments.","marker":"[31]"},{"why":"Supplies the trace bound used to replace the non-convex MSD constraint in the sampling optimization with a convex relaxation.","marker":"[42]"},{"why":"Supplies the generalized-concavity result used to argue that the sub-level sets of the relaxed sampling problem are convex.","marker":"[43]"},{"why":"Supplies the adapt-then-combine diffusion recursion that the distributed Topo-LMS algorithm is built on.","marker":"[46]"},{"why":"Supplies the real traffic-measurement dataset used to evaluate the centralized and distributed algorithms against the graph baseline.","marker":"[48]"}],"fun_headline_variants":["Topological LMS: adaptive edge-flow tracking from partial samples","Convergence and MSE bounds for LMS on simplicial complexes","Distributed topological LMS beats graph-based adaptive filters"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 3: the sampled regressors $D(n)X(n)$ are treated as temporally white and independent across time, even though the model builds them from delayed copies of the same signal $x(n)$, so the analysis's closed-form MSD and convergence-rate formulas are only exact under a condition the model itself violates.","fun_headline_variants_meta":{"raw":{"variants":["Topological LMS: adaptive edge-flow tracking from partial samples","Convergence and MSE bounds for LMS on simplicial complexes","Distributed topological LMS beats graph-based adaptive filters"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000561,"raw_usage":{"total_tokens":2704,"prompt_tokens":1023,"completion_tokens":1681,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":639,"completion_tokens_details":{"reasoning_tokens":1636}},"tokens_in":639,"tokens_out":1681,"duration_ms":14362,"temperature":1.0,"reasoning_tokens":1636,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:52:03.467049+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Topo-LMS on data generated by the paper's model (8) with strongly autocorrelated input, e.g. an AR(1) process with coefficient close to one, and compare measured steady-state MSD and convergence rate with formulas (32) and (34); a gap that grows with the autocorrelation would show those closed-form expressions depend essentially on temporal whiteness.","supporting_citations":[{"cited_title":"Topological signal processing over simplicial complexes,","cited_arxiv_id":null,"evidence_quote":"Supplies the Hodge Laplacian definitions, simplicial Fourier transform, and topology-inference tools that define $L_u$ and $L_d$ in the observation model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the LMS stochastic-analysis machinery, including the independent-regressor assumption and weighted variance recursions used in Section III."},{"cited_title":"Adaptation, learning, and optimization over networks,","cited_arxiv_id":null,"evidence_quote":"Supplies the distributed adaptation and network-optimization framework used for the stability analysis of the distributed variant."},{"cited_title":"Adaptive graph signal processing: Algorithms and optimal sampling strategies,","cited_arxiv_id":null,"evidence_quote":"Supplies the graph-domain adaptive LMS with optimal sampling strategies that this paper generalizes to simplicial complexes."},{"cited_title":"Distributed diffusion adaptation over graph signals,","cited_arxiv_id":null,"evidence_quote":"Supplies the graph-based LMS model and distributed diffusion adaptation that Topo-LMS extends and compares against in the experiments."},{"cited_title":"Trace bounds on the solution of the algebraic matrix riccati and lyapunov equation,","cited_arxiv_id":null,"evidence_quote":"Supplies the trace bound used to replace the non-convex MSD constraint in the sampling optimization with a convex relaxation."},{"cited_title":"Avriel, W","cited_arxiv_id":null,"evidence_quote":"Supplies the generalized-concavity result used to argue that the sub-level sets of the relaxed sampling problem are convex."},{"cited_title":"Diffusion LMS strategies for distributed estimation,","cited_arxiv_id":null,"evidence_quote":"Supplies the adapt-then-combine diffusion recursion that the distributed Topo-LMS algorithm is built on."},{"cited_title":"Sndlib 1.0—survivable network design library,","cited_arxiv_id":null,"evidence_quote":"Supplies the real traffic-measurement dataset used to evaluate the centralized and distributed algorithms against the graph baseline."}],"review_version":1}