Transient Error Analysis of the LMS and RLS Algorithm for Graph Signal Estimation

Chengjin Li; Haiquan Zhao

REVIEW 3 major objections 5 minor 25 references

Transient Error Analysis of the LMS and RLS Algorithm for Graph Signal Estimation

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper derives closed-form time-dependent mean-square deviation formulas for the GSP LMS and GSP RLS adaptive graph-signal estimators under Gaussian noise, fixed spectral parameter, and fixed sampling set.

desk verdict The new transient MSD formulas for GSP LMS/RLS rest on an invalid collapse of the noise history into a single w(t), so the main result does not follow. read the letter →

arxiv 2506.00403 v1 pith:TCEBQDIZ submitted 2025-05-31 eess.SP

Haiquan Zhao , Chengjin Li This is my paper · ORCID

classification eess.SP

keywords graphsignalprocessingleastmeansquarerecursivesquarestransienterroranalysisdeviationestimationGaussiannoise

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

This paper tries to establish closed-form, time-dependent formulas for the transient mean-square deviation (MSD) of the two main adaptive graph-signal estimators: GSP LMS and GSP RLS. Until now, only steady-state error analysis existed for these algorithms, so engineers could not predict how fast or how accurately the estimates converge at intermediate time steps. Under Gaussian noise with known covariance, a fixed spectrally sparse parameter F, and a fixed sampling set, the paper derives GMSD(t) expressions by expanding the frequency-domain error recursion and summing the resulting matrix-geometric series. If the derivation is correct, it gives a quantitative tool for predicting and tuning the convergence behavior of graph adaptive filters before they reach steady state.

What carries the argument

The carrying object is the graph-frequency error recursion. For GSP LMS, $\Delta\hat{\mathbf{s}}_F(t+1) = \mathbf{A}\Delta\hat{\mathbf{s}}_F(t) + \mu\mathbf{U}_F^T\mathbf{D}_S\mathbf{w}(t)$ with $\mathbf{A}=\mathbf{I}-\mu\mathbf{U}_F^T\mathbf{D}_S\mathbf{U}_F$; for GSP RLS, $\Delta\hat{\mathbf{s}}_F(t+1) = \lambda\Delta\hat{\mathbf{s}}_F(t) + (1-\lambda)\mathbf{M}\mathbf{U}_F^T\mathbf{D}_S\mathbf{C}_w^{-1}\mathbf{w}(t)$. Mathematical induction expands the recursion into a sum of matrix powers applied to past noise terms, and the paper evaluates the sum with the geometric-series identity involving $(\mathbf{I}-\mathbf{A})^{-1}$. The independence assumptions A1–A5 let the expectation of the squared error factor into terms involving the noise covariance $\mathbf{C}_w$, producing the closed-form $\mathrm{GMSD}(t)$ expressions.

What would settle it

Recompute the transient expectation without collapsing the sum $\sum_{j=0}^{t-1}\mathbf{A}^{t-1-j}\mathbf{U}_F^T\mathbf{D}_S\mathbf{w}(j)$ into a single $\mathbf{w}(t)$ term, and compare the resulting $\mathrm{GMSD}(t)$ with equation (17); any difference at early times settles the matter. Concretely, simulate GSP LMS with white Gaussian noise ($\mathbf{C}_w=\sigma^2\mathbf{I}$), average 50 or more runs, and check whether the empirical MSD matches equation (17) at $t=2,3,4$ or instead follows the unreduced sum.

Watch

Extended reading notes

Core claim

The paper's central claim is that the transient estimation error of the GSP LMS and GSP RLS algorithms can be expressed in closed form at every time instant $t$. Working in the graph-frequency domain, the paper represents the error $\Delta \hat{\mathbf{s}}_F(t)$ as a sum of powers of the update matrix $\mathbf{A} = \mathbf{I} - \mu \mathbf{U}_F^T \mathbf{D}_S \mathbf{U}_F$ acting on the initial error plus an accumulated noise term, and similarly for RLS with $\lambda$ and $\mathbf{M}$. Using the matrix geometric-series identity $(\mathbf{I} - \mathbf{A})^{-1}$, it converts that sum into formulas (17) and (22) for $\mathrm{GMSD}(t)$ that depend only on the noise covariance $\mathbf{C}_w$, the graph structure $\mathbf{U}_F$, the sampling matrix $\mathbf{D}_S$, the step size $\mu$ (or forgetting factor $\lambda$), and $t$. Simulations on the Brazilian temperature dataset with three Gaussian noise settings are presented as verification.

Load-bearing premise

The formulas hinge on treating the whole history of noise samples as a single current noise vector when summing powers of the update matrix; if that step is not valid for the independent noise used in the experiments, the closed-form transient error is not the true expectation.

Editorial extensions

If this is right

GSP LMS and GSP RLS convergence can now be predicted quantitatively at every iteration, not just in steady state, for the fixed-graph, known-covariance setting.
The formulas expose how step size $\mu$ and forgetting factor $\lambda$ shape the transient trajectory, so parameter choices can be tuned to a target convergence time or overshoot.
Because the expressions depend only on second-order noise statistics, the analysis covers the Gaussian scenarios tested and gives a benchmark for comparing graph adaptive filters.
Simulated MSD curves can be checked against equations (17) and (22) for other graphs, sampling sets, and noise covariance matrices without rerunning the derivation.

Reading between the lines

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

Editorial inference: the same induction-plus-geometric-sum argument should extend to any noise process with finite covariance matrix, since the final formulas rely on $\mathbf{C}_w$ rather than on Gaussianity itself; the paper's Gaussian assumption may be stronger than its algebra requires.
Editorial inference: differentiating $\mathrm{GMSD}(t)$ with respect to $t$ or to $\mu$ could yield closed-form convergence-rate and optimal-step-size conditions that the paper does not state.
Editorial inference: the paper's own closing remark that time-varying $F$ and sampling set remain future work is a real limit: if $\mathbf{U}_F$ changes between iterations, the geometric-series collapse at the heart of equations (17) and (22) no longer applies.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The paper targets a real gap: closed-form transient mean-square-error expressions for LMS and RLS on graph signals. Prior work covers steady state only, and the authors correctly note that no one has published the time-dependent GMSD(t) formulas. The topic is relevant, the setup follows the literature cleanly, and the simulations use a real temperature dataset with several graph configurations. Credit where due: the problem selection is good, and the paper is honest about its scope.

Unfortunately, the central derivation is wrong. In Section III-A, Eq. (11) writes the noise term as μ[A^{t-2} + ... + I] U_F^T D_S w(t), which would be correct only if the noise vector were the same at every time step. The exact recursion (7) gives a convolution over all past noises: μ Σ_{j=1}^{t-1} A^{t-1-j} U_F^T D_S w(j). The geometric-series trick in Eqs. (12)-(16) solves for a surrogate K that contains w(t) in every term, not the true sum. Under the stated zero-mean Gaussian i.i.d. assumption, the covariance of the true noise term is a sum of t-1 distinct contributions, not the single C_w block that appears in Eq. (17). The same collapse appears in the RLS derivation: the induction from (18)-(21) drops the λ-weighted history and keeps only the latest noise sample. So neither GMSD expression follows from the stated model, and the 50-run ensemble averages cannot validate formulas derived from a different, effectively constant-noise process.

This is not a minor slip; the load-bearing algebra is invalid. The paper would be a genuine contribution if the derivation were repaired, because the intended result is useful for convergence-speed prediction and step-size design in graph adaptive filtering. As written, I would not cite it as a source for transient formulas. The authors could plausibly fix it by carrying the full noise sum and evaluating its covariance properly; the final expressions will be more involved but still derivable.

The appropriate reader is someone working on graph adaptive filter theory who needs to know this gap remains open. For peer review, a serious editor could send it out because the topic is important and the error is subtle enough to require expert judgment, but the verdict should be rejection rather than acceptance. The paper deserves a serious referee, but not publication in its current form.

Regards,

Referee Report

3 major / 5 minor

Summary. The paper aims to derive closed-form transient mean-square-deviation (GMSD) expressions for the graph-signal LMS and RLS algorithms under Gaussian noise, and to validate these expressions with simulations on the Brazilian temperature dataset. The derivation proceeds by writing the frequency-domain estimation error recursion, applying mathematical induction, and then taking expectations under the stated Gaussian noise covariance. The central claim is that Eqs. (17) and (22) give the time-dependent GMSD for the two algorithms, and that Figs. 3 and 4 demonstrate agreement with 50-run ensemble simulations.

Significance. If correct, the paper would supply a genuinely useful quantitative tool for predicting the transient behavior of graph adaptive filters, complementing existing steady-state analyses. The paper has some praiseworthy features: the theoretical curves are not fitted to the simulations, the noise covariance and algorithm parameters are set independently, and the problem of transient analysis for GSP algorithms is well motivated. However, the central derivation contains a load-bearing error: the convolution over the past noise samples is collapsed into a single noise vector, which is not valid under the stated i.i.d. Gaussian model. Because both claimed GMSD formulas rest on this collapse, the main contribution is not established, and the simulation agreement cannot be interpreted as validation of the proposed theory.

major comments (3)

[Section III-A, Eqs. (11)-(16)] The transient LMS derivation replaces the weighted sum of distinct past noise samples by a single vector w(t). Starting from the recursion (7), the exact solution is Δs_F(t)=A^{t-1}Δs_F(1)+μ Σ_{j=1}^{t-1} A^{t-1-j} U_F^T D_S w(j). In Eqs. (12)-(13), however, every term of K uses the same w(t), which would be correct only if w(1)=...=w(t). This contradicts assumption A3 and the i.i.d. Gaussian noise used in the simulations. Consequently, the subtraction in (14), the inversion in (15), and the resulting expression (16) solve for a surrogate quantity rather than for the actual weighted noise sum. Under A3, the correct noise covariance is μ^2 Σ_{j=1}^{t-1} A^{t-1-j} U_F^T D_S C_w D_S U_F (A^{t-1-j})^T, a sum of t-1 distinct covariance contributions, not the single C_w block appearing in (17). Therefore Eq. (17) is not the expected GMSD under the stated model.
[Section III-B, Eqs. (18)-(22)] The RLS derivation contains the same error. The induction from (18)-(20) to (21) keeps only the latest noise term w(t) and drops the λ^{t-1-j}-weighted history of past noise samples. The exact solution of (18) is Δs_F(t)=λ^{t-1}Δs_F(1)+(1-λ)Σ_{j=1}^{t-1} λ^{t-1-j} M U_F^T D_S C_w^{-1} w(j), and the covariance of this term is not the single covariance expression used in (22). As a result, Eq. (22) does not give the expected GMSD for the i.i.d. Gaussian noise process described in Section II and used in the experiments.
[Section IV, Figs. 3 and 4] Because Eqs. (17) and (22) are derived under a different, effectively constant-noise process, the visual agreement reported in Figs. 3 and 4 cannot validate the proposed theoretical analysis. The manuscript reports no quantitative discrepancy measure, such as relative error, mean absolute error, or confidence intervals for the 50-run averages, so the claim of a 'better match' is not supported by any statistic. The simulations would need to be compared against the corrected transient expressions to test the theory.

minor comments (5)

[Section II, Eq. (5)] The notation for the performance metric is inconsistent: the text defines GMSD(t) in (5), but Eq. (6) and later equations use Δs_F(t) without a clear definition, and Eq. (23) plots a version scaled by 10 log10. Please define all symbols precisely and state whether the figures plot dB values.
[Section IV, Eq. (24)] The definition of C_w in (24) is ambiguous: a is described as a vector whose components satisfy a Gaussian distribution, while diag(aN_a+bN_b) suggests a deterministic diagonal matrix. Please clarify whether a is a random vector realization or a parameter, and ensure that the resulting covariance matrix is positive definite in all three scenarios.
[Section III, Eqs. (12)-(17)] The equations in Section III contain numerous typographical and formatting problems (missing subscripts, misplaced matrix products, garbled diag and sqrt terms). These need a careful rewrite, independent of the technical issue above, because they make the derivation difficult to follow.
[Section IV, Cases I and II] The experimental setup specifies K, F, and |S|, but not which nodes are sampled or how the graph shift/adjacency matrix is constructed from the temperature stations. This information is needed for reproducibility and for checking that the constants in the theory match the simulations.
[Section V, Conclusion] The conclusion lists only time-varying F and sampling set as future work, but the manuscript also relies on the Gaussian noise assumption and on the independence assumption A2, whose role in the derivation should be discussed more carefully.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the transient GMSD derivations follow (or attempt to follow) from the algorithm recursions and the stated noise covariance, with no fitted parameters or load-bearing self-citations.

full rationale

The paper's central product is a closed-form transient GMSD expression for GSP LMS and GSP RLS. The derivation starts from the published update recursions in Eqs. (3)-(4), defines the frequency-domain error Δs_F(t), and under stated assumptions A1-A5 (including Gaussian noise with covariance C_w) derives E[||Δs_F(t)||^2]. The theoretical curves are evaluated using the same user-set step sizes/forgetting factors and the same noise covariance used to generate the simulated noise; this is standard model evaluation, not fitting the predicted curve to the measured curve. No parameter is calibrated to make the theory match the simulations. The paper's citations to [11], [19], [21], [22] provide background or prior algorithm definitions; the transient-error induction in Section III does not reduce to any of those cited results, and no uniqueness or existence claim is imported from the authors' own prior work to force the chosen formulas. The obvious weakness in the manuscript is a derivation error, not circularity: in Eq. (12) the authors collapse the history of i.i.d. noise samples w(1),...,w(t) into a single w(t), so the geometric-series step (14) does not follow under A3 and the resulting GMSD(t) in Eqs. (17) and (22) is not the expectation of the true transient error. That is a correctness or derivability defect, not a self-referential reduction: the intended derivation is genuinely attempting to compute a new quantity from independent inputs. The conclusion also honestly records that the analysis is limited to fixed F and S. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No new entities or fitted constants are introduced. The step sizes and forgetting factors are user-set algorithm hyperparameters. The critical unstated premise is that the noise sequence collapses to a single sample in the transient sum, which is invalid.

assumptions (6)

domain assumption A1: The graph signal is bandlimited or sparse in the graph structure.
Used in Section III to justify working with the low-dimensional frequency representation s_F and the graph structure matrix U_F.
domain assumption A2: U_F, Delta(s_F)(t), and s_F are independent of each other.
Invoked when taking expectations in Eqs. (17) and (22). Since Delta(s_F)(t) depends on past noise, this independence is nontrivial and not justified.
domain assumption A3: w(t) is zero-mean Gaussian noise with covariance E[w(t) w(t)^T] = C_w.
Used to evaluate the mean-square expectation of the error expressions.
domain assumption A4: F is fixed, so U_F is a constant matrix.
Stated in Section III and used throughout the derivation.
domain assumption A5: S is fixed, so the sampling matrix D_S is constant.
Stated in Section III and used throughout the derivation.
ad hoc to paper Implicit assumption: noise samples at different time instants are interchangeable, so w(2)=...=w(t) when summing the transient terms.
Used in Eqs. (12)-(16) to telescope the sum into a single geometric-series term. This is false for the i.i.d. Gaussian noise used in the experiments.

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Cite this review

Pith. "Pith review of Transient Error Analysis of the LMS and RLS Algorithm for Graph Signal Estimation." pith.science (2026). https://pith.science/paper/TCEBQDIZ

@misc{pith2026250600403,
  author       = {Pith},
  title        = {Pith review of: Transient Error Analysis of the LMS and RLS Algorithm for Graph Signal Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TCEBQDIZ}},
  note         = {Machine review of arXiv:2506.00403}
}

read the original abstract

Recently, the proposal of the least mean square (LMS) and recursive least squares (RLS) algorithm for graph signal processing (GSP) provides excellent solutions for processing signals defined on irregular structures such as sensor networks. The existing work has completed the steady state error analysis of the GSP LMS algorithm and GSP RLS algorithm in Gaussian noise scenarios, and a range of values for the step size of the GSP LMS algorithm has also been given. Meanwhile, the transient error analysis of the GSP LMS algorithm and GSP RLS algorithm is also important and challenging. Completing the above work will help to quantitatively analyze the performance of the graph signal adaptive estimation algorithm at transient moments, which is what this paper is working on. By using formula derivation and mathematical induction, the transient errors expressions of the GSP LMS and GSP RLS algorithm are given in this paper. Based on the Brazilian temperature datasets, the related simulation experiments are executed, which strongly demonstrate the correctness of our proposed theoretical analysis

Figures

Figures reproduced from arXiv: 2506.00403 by the authors.

**Figure 2.** Graph based on July data in Brazil dataset(Case II) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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