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REVIEW 3 major objections 4 minor 49 references

Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Solid incremental extension of graph-LMS to simplicial complexes; the analysis is standard, the caveat is admitted, and the paper deserves a serious referee. read the letter →

arxiv 2505.23160 v1 pith:IWH6775Q submitted 2025-05-29 eess.SP cs.LG

Lorenzo Marinucci , Claudio Battiloro , Paolo Di Lorenzo This is my paper
classification eess.SPcs.LG
keywords adaptivefilteringleastmeansquaressimplicialcomplexestopologicalsignalprocessingHodgeLaplacianedgeflowpredictionsamplingstrategiesdistributeddiffusion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Sec. III.B, Assumption 3, Eqs. (8)-(9)] 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.
  2. [Sec. III.B and Appendix A, Eqs. (29)-(32), (62)-(64)] 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.
  3. [Sec. III.C, Eqs. (35)-(37) and Sec. VI.B] 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.
minor comments (4)
  1. [Notation paragraph] 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.
  2. [Appendix B] 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.
  3. [Sec. VI.A and Sec. VI.D] 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'.
  4. [Sec. III.C, Eq. (37)] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the LMS analysis and MSD formulas are derived from stated assumptions rather than fitted to data, and the traffic-data comparison is external; the minor self-reference is the preliminary-version citation [1] and a synthetic sampling validation that reuses the same design model.

full rationale

The derivation chain is self-contained. Theorem 1 (Sec. III.B-C, Appendix A) starts from the Topo-LMS recursion (19), defines the error recursion (21), and under Assumptions 1-3 obtains (24), (26), (29), and then (31), from which (32) and (34) follow by standard Kronecker-product expansions; these are consequences of the algorithm's definition, not restatements of an assumed conclusion. The optimal sampling problem (35)-(37) is an optimization over the derived MSD and rate expressions, and Sec. VI.B validates it by generating data from the same model (8) with the optimized probabilities; this is a self-consistent design check, not a prediction reduced to a fit. The real-data experiments (Sec. VI.A and VI.D) compare against a graph-only Edge-Laplacian LMS baseline on held-out traffic snapshots, so the performance claim has independent empirical content. The paper's citations to the authors' own preliminary work [1] and related TSP papers are motivational and background; no uniqueness theorem or ansatz is imported from [1] to force the conclusion. The one admitted limitation is Assumption 3 (Sec. III.B): 'Even if Assumption 3 is not strictly verified in model (8)', and the analogous Assumption 4 in Sec. V.A; this means the closed-form MSD and convergence rate are heuristic under the generative model (9), because consecutive regressors share delayed signals. That is a correctness and assumption gap, not a circular step: the formulas are not equivalent to their inputs by construction. Accordingly, no specific circular reduction meeting the quote-and-exhibit standard was found; the score reflects only the minor, non-load-bearing self-citation and the self-referential synthetic sampling validation.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The theoretical results rest primarily on the standard LMS analysis toolbox: stationarity and independent-sampling assumptions to compute statistical moments, the independent-regressor assumption (which the paper admits is violated by its own model), and small-step-size approximations to obtain closed-form MSD expressions. The topology-inference section additionally relies on a proximal hard-thresholding operator and on hand-selected regularization parameters. No new physical entities are introduced.

free parameters (3)
  • Step-size μ = 10^-2 (centralized synthetic), 10^-4 and 10^-1 (real data), 10^-2 (distributed)
    Chosen by hand per experiment to satisfy stability condition (25); not fitted to force the reported curves.
  • Regularization thresholds λ0, λ1 = 0.1
    Hand-selected in topology-inference experiments to satisfy 1 - sqrt(2λ1) > sqrt(2λ0); control the sparsity/activation trade-off in (41)-(44).
  • Filter order M = M = 2 or 3 in experiments
    Model-order hyperparameter chosen by hand for each dataset; the theory holds for any M ≥ 1.
assumptions (8)
  • domain assumption Assumption 1: the sampling variables d_i(l) are spatially and temporally independent.
    Used to compute expectations like E{x^T (L^m)^T D(n) L^l x} = Tr((L^m)^T P L^l Cx(m-l)); standard in adaptive filtering with random sampling.
  • domain assumption Assumption 2: the signal x(n) is zero-mean wide-sense stationary.
    Makes CX and cXy time-invariant and the normal equations well-defined; invoked in Eq. (11)-(16).
  • ad hoc to paper Assumption 3: D(n)X(n) is temporally white with covariance CX.
    The paper admits this is 'not strictly verified in model (8)' because X(n) contains delayed copies of x(n), yet the entire mean-square analysis, the F≈Q^T⊗Q^T substitution, and Theorem 1 rely on it.
  • domain assumption Assumption 4: local regressors zi(n) are temporally white.
    Standard in diffusion-LMS analysis; used to derive the mean recursion (53) and mean-square recursion (56) for Algorithm 2.
  • domain assumption Small step-size approximation: F=E{Q^T(n)⊗Q^T(n)} ≈ Q^T⊗Q^T, neglecting higher-order powers of μ.
    Standard in LMS literature when μ is small; used to obtain the closed-form MSD (31) and the convergence-rate approximation (34).
  • domain assumption The observed edge signal is generated by the simplicial FIR model (8) with known Laplacians (or estimated upper Laplacian for topology inference).
    The entire estimation problem is defined relative to this generative model; if the true data generation differs, the optimal filter coefficients ho are undefined.
  • standard math Lemma 2 matrix expansion (I - μZ)^{-1} = I + μR(μ) with bounded R(μ).
    Uses the Neumann series for 0 < μ < 1/λmax(Z); applied in Appendix A to bound the O(μ^2) term in Theorem 1.
  • ad hoc to paper Convexity/pseudo-convexity of the relaxed sampling problem (37).
    The authors assert constraint (c) defines a pseudo-convex map with convex sublevel sets, citing [43], but no proof is given in the paper; the practical solvability of (37) depends on this.

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Cite this review

Pith. "Pith review of Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes." pith.science (2026). https://pith.science/paper/IWH6775Q

@misc{pith2026250523160,
  author       = {Pith},
  title        = {Pith review of: Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWH6775Q}},
  note         = {Machine review of arXiv:2505.23160}
}
read the original abstract

This paper introduces a novel adaptive framework for processing dynamic flow signals over simplicial complexes, extending classical least-mean-squares (LMS) methods to high-order topological domains. Building on discrete Hodge theory, we present a topological LMS algorithm that efficiently processes streaming signals observed over time-varying edge subsets. We provide a detailed stochastic analysis of the algorithm, deriving its stability conditions, steady-state mean-square-error, and convergence speed, while exploring the impact of edge sampling on performance. We also propose strategies to design optimal edge sampling probabilities, minimizing rate while ensuring desired estimation accuracy. Assuming partial knowledge of the complex structure (e.g., the underlying graph), we introduce an adaptive topology inference method that integrates with the proposed LMS framework. Additionally, we propose a distributed version of the algorithm and analyze its stability and mean-square-error properties. Empirical results on synthetic and real-world traffic data demonstrate that our approach, in both centralized and distributed settings, outperforms graph-based LMS methods by leveraging higher-order topological features.

Figures

Figures reproduced from arXiv: 2505.23160 by the authors.

Figure 1
Figure 1. (Left) The edge (1, 4) is lower-adjacent to the blue edges (denoted with N (d) (1,4)). (Right) The edge (1, 4) is upper￾adjacent to the red edges (denoted with N (u) (1,4)). Hodge Laplacians. From the incidence information, it is possible to define the Hodge Laplacian matrices as follows. L0 = B1B T 1 Lk = B T k Bk + Bk+1B T k+1, k = 1, ..., K − 1 (4) LK = B T KBK Note that each Laplacian of intermediate order, k = … view at source ↗
Figure 2
Figure 2. MSD vs iteration index, for different percentages of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. (Top) Prediction error over training data. (Bottom) Performance comparison of different methods over test data. over 30 realizations, along with the corresponding theoretical steady-state values in (31)) depicted as dashed horizontal lines. As observed, the simulation results closely align with the theoretical predictions. We designed the test such that, for each sampling percentage X, the X-th percentile of edges w… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Optimal sampling probabilities, obtained as solution of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: MSD’s evolution over time with varying convergence [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 7
Figure 7. Figure 7: (Top) Filter estimation’s error with time-varying topol￾ogy of the simplicial complex. (Bottom) Topology estimation’s error with time-varying topology of the simplicial complex. from {10−6 , 10−4 , 10−3 , 10−2}. We conduct three simulations, one for each simplicial com…
Figure 9
Figure 9. Figure 9: Performance comparison over test data of the Topo-LMS [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

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    Moreover, it holds that: ∥I−µC X ∥2 2 =max{(1−µδ) 2,(1−µν) 2} ≤1−2µδ+µ 2ν2 = 1−2µδ 1− µν2 2δ (a) ≈1−2µδ where (a) holds when µ≪2δ/ν 2. This proves (34), and concludes the proof of Theorem 1. APPENDIXB PROOFOFLEMMA1 Since problem (43) is separable in each coordinate ui of u, we...

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