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arxiv: 2604.26787 · v1 · submitted 2026-04-29 · 💻 cs.LG · eess.SP

Hankel and Toeplitz Rank-1 Decomposition of Arbitrary Matrices with Applications to Signal Direction-of-Arrival Estimation

Georgios I. Orfanidis , Dimitris A. Pados , George Sklivanitis , Elizabeth Serena Bentley This is my paper

Pith reviewed 2026-05-07 13:26 UTC · model grok-4.3

classification 💻 cs.LG eess.SP
keywords Hankel matrixToeplitz matrixrank-1 approximationdirection of arrival estimationmaximum likelihoodL2 normL1 normfew-shot estimation
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The pith

Optimal rank-1 Hankel and Toeplitz approximations under L2 and L1 norms produce maximum-likelihood DoA estimators for few-shot scenarios.

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

The paper presents algorithms to compute the best rank-1 approximation of any matrix when the approximation must have Hankel or Toeplitz structure, using either squared-error or absolute-error measures. These decompositions are then applied to create direction-of-arrival estimators that require only a small number of signal samples. The estimators are proven to be maximum likelihood when the noise follows a white Gaussian distribution for the L2 case and a Laplace distribution for the L1 case. This matters for autonomous systems and sensing applications where data is limited and noise may not be Gaussian. The methods are tested in simulations and on real data to confirm their effectiveness.

Core claim

We consider the problems of computing the optimal rank-1 Hankel and Toeplitz-structured approximation of arbitrary matrices under L2 and L1-norm error. Such problems arise naturally in engineered systems, including the basic few-shot signal Direction-of-Arrival (DoA) estimation problem that is of importance to modern autonomous systems applications. We develop accurate and computationally efficient structured matrix decomposition algorithms for both formulations and then derive analytically grounded small-sample-support DoA estimators for practical sensing system deployments. The resulting estimators under the L2 and L1 norms are formally shown to be maximum-likelihood optimal under white G

What carries the argument

Rank-1 Hankel and Toeplitz structured matrix decomposition under L2 and L1 norms, which extracts the signal component while respecting the structure induced by the array geometry or time series.

If this is right

  • DoA estimators become maximum-likelihood optimal under matching noise distributions.
  • Performance improves in few-shot regimes with limited samples.
  • Applicable to both Gaussian and heavy-tailed Laplace noise models.
  • Computationally efficient algorithms allow real-time use in autonomous systems.
  • Validated on both simulated and real-world data for reliability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The decomposition technique could be adapted for other structured approximations in array signal processing.
  • It separates the signal structure from the noise statistics, allowing flexible noise modeling.
  • In practice, this might lead to better integration with adaptive sensing systems that adjust to observed noise.
  • Extensions to higher-rank structures or multi-dimensional arrays could follow similar principles.

Load-bearing premise

The observed matrix is a noisy version of an exact rank-1 matrix with Hankel or Toeplitz structure generated by the signal model.

What would settle it

Running the proposed estimators on data from a known DoA scenario with added white Gaussian noise and comparing their accuracy to the theoretical maximum-likelihood bound or to unstructured estimators would show if they achieve the claimed optimality and superiority in small-sample cases.

Figures

Figures reproduced from arXiv: 2604.26787 by Dimitris A. Pados, Elizabeth Serena Bentley, George Sklivanitis, Georgios I. Orfanidis.

Figure 1
Figure 1. Figure 1: Algorithm for complex rank-1 Hankel approximation of view at source ↗
Figure 2
Figure 2. Figure 2: Algorithm for complex rank-1 Hankel approximation of view at source ↗
Figure 3
Figure 3. Figure 3: Sliding-window acquisition over a uniform linear array (ULA) view at source ↗
Figure 4
Figure 4. Figure 4: Mean absolute estimation error versus number of antenna array elements: (a) SNR = -5 dB, (b) SNR = 0 dB, (c) SNR = 5 dB, (d) view at source ↗
Figure 5
Figure 5. Figure 5: Mean absolute estimation error versus number of antenna array elements, impulse probability view at source ↗
Figure 6
Figure 6. Figure 6: Mean absolute estimation error versus number of antenna array elements, impulse probability view at source ↗
Figure 7
Figure 7. Figure 7: Sliding-window acquisition over a 5 × 8 URA with D = 4 available RF processing chains. ‘X’ markers indicate unavailable antenna element measurements, which lead to an irregular effective array geometry and corrupted sliding-window data [50]. Appendix Proof of Theorem 1 Proof: Following the developments in (11)–(13) we rewrite ˆθL2 = argmin θ∈[−90◦,90◦) view at source ↗
Figure 8
Figure 8. Figure 8: Mean absolute DoA estimation error for UAV hovering view at source ↗
read the original abstract

We consider the problems of computing the optimal rank-$1$ Hankel and Toeplitz-structured approximation of arbitrary matrices under $L_2$ and $L_1$-norm error. Such problems arise naturally in engineered systems, including the basic few-shot signal Direction-of-Arrival (DoA) estimation problem that is of importance to modern autonomous systems applications. We develop accurate and computationally efficient structured matrix decomposition algorithms for both formulations and then derive analytically grounded small-sample-support DoA estimators for practical sensing system deployments. The resulting estimators under the $L_2$ and $L_1$ norms are formally shown to be maximum-likelihood optimal under white Gaussian and Laplace noise, respectively. The estimators are further validated through extensive simulation studies and real-world data experiments in few-shot DoA inference.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper develops algorithms for computing the optimal rank-1 Hankel and Toeplitz structured approximations to arbitrary matrices under L2 and L1 norms. It applies these to derive small-sample-support DoA estimators and analytically shows that the L2-based estimator is maximum-likelihood optimal under white Gaussian noise while the L1-based estimator is optimal under Laplace noise. The claims are supported by algorithmic descriptions, theoretical derivations, simulations, and real-world experiments.

Significance. If the ML optimality claims hold under the physical sensing model, the work would provide a significant contribution by delivering parameter-free, theoretically grounded DoA estimators suitable for few-shot regimes in autonomous systems. The structured decomposition algorithms may also have broader utility in signal processing applications involving Hankel or Toeplitz matrices. The paper merits credit for combining analytical derivations with extensive empirical validation on both simulated and real data.

major comments (1)
  1. [DoA application and ML optimality derivation] § on DoA signal model and ML derivation (the section deriving optimality from the likelihood): The claim that the L2 (resp. L1) rank-1 Hankel/Toeplitz approximant is ML optimal under white Gaussian (Laplace) noise assumes iid noise across the observed matrix entries. However, the motivating DoA model forms the matrix via Hankel/Toeplitz embedding of the short vector y = a(θ)s + n where n is white vector noise; overlapping entries therefore contain identical noise samples and the effective noise on the matrix is correlated. This mismatch means the formal optimality proof applies to an observation model that does not match the physical sensing model used to motivate the estimators. This is load-bearing for the central claim in the abstract.
minor comments (2)
  1. [Abstract] The abstract asserts that the estimators are 'formally shown' to be ML optimal but does not cite the specific theorem, equation, or subsection containing the derivation, which reduces readability.
  2. [Problem formulation] Notation for the embedding operators that construct the Hankel and Toeplitz matrices from the vector y should be introduced with an explicit definition and example in the problem formulation section to aid clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting this important distinction between the matrix-level noise model used in the optimality derivation and the vector-level white noise model that motivates the DoA application. We address the comment below.

read point-by-point responses

  1. Referee: [DoA application and ML optimality derivation] § on DoA signal model and ML derivation (the section deriving optimality from the likelihood): The claim that the L2 (resp. L1) rank-1 Hankel/Toeplitz approximant is ML optimal under white Gaussian (Laplace) noise assumes iid noise across the observed matrix entries. However, the motivating DoA model forms the matrix via Hankel/Toeplitz embedding of the short vector y = a(θ)s + n where n is white vector noise; overlapping entries therefore contain identical noise samples and the effective noise on the matrix is correlated. This mismatch means the formal optimality proof applies to an observation model that does not match the physical sensing model used to motivate the estimators. This is load-bearing for the central claim in the abstract.

    Authors: We agree that the formal ML optimality derivation in the paper is performed under the assumption of i.i.d. noise across the entries of the observed matrix. In the DoA section, the observed matrix is obtained by Hankel/Toeplitz embedding of a short vector y whose noise component is white (i.i.d. across vector entries). Because of the repeated entries in the embedding, the induced noise on the matrix entries is correlated and the Frobenius (or element-wise L1) distance on the matrix is not equivalent to the standard Euclidean (or L1) distance on the original vector. Consequently, the rank-1 structured approximant is exactly maximum-likelihood only for the matrix observation model with i.i.d. entries; it is not strictly ML for the vector white-noise model that physically motivates the DoA problem. We will revise the manuscript as follows: (i) explicitly state the noise model under which the optimality holds, (ii) qualify the abstract claim to indicate that the estimators are ML-optimal under the matrix i.i.d. model and are motivated by (but not exactly optimal for) the embedded vector model, and (iii) add a short discussion of the relationship between the two models together with a note that the matrix-norm formulation remains a principled and parameter-free estimator for few-shot DoA. These changes directly address the load-bearing concern for the central claim. revision: yes

Circularity Check

0 steps flagged

No circularity: ML optimality derived from standard likelihood under explicit iid matrix noise model

full rationale

The paper derives the L2 and L1 rank-1 Hankel/Toeplitz approximants analytically and states that they are ML optimal under white Gaussian and Laplace noise, respectively. This follows directly from the standard log-likelihood for iid entries (Gaussian or Laplace) plus a rank-1 structured signal; the derivation does not reduce to a fitted parameter renamed as prediction, nor to any self-citation chain, nor to a self-definitional loop. The DoA application simply instantiates the same structured-matrix model; any mismatch between vector noise and induced matrix-entry correlations is an external modeling assumption, not a circular reduction inside the paper's equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions in signal processing about matrix structure from single-source models and noise distributions. No free parameters or new entities are mentioned in the abstract.

axioms (2)
  • domain assumption The underlying signal model results in a rank-1 Hankel or Toeplitz matrix structure.
    This is the basis for the approximation problem in DoA estimation.
  • domain assumption Noise is white (independent and identically distributed) Gaussian for L2 and Laplace for L1.
    Required for the maximum-likelihood optimality claims.

pith-pipeline@v0.9.0 · 5453 in / 1340 out tokens · 71947 ms · 2026-05-07T13:26:09.761833+00:00 · methodology

discussion (0)

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