arxiv: 2605.06198 · v1 · submitted 2026-05-07 · 📡 eess.SP

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Orthogonal Least Squares with Integrated Information Theoretic Criteria for Joint Number of Targets and DoA Estimation

Martin Willame , Gilles Monnoyer , Fran\c{c}ois Horlin , J\'er\^ome Louveaux

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Pith reviewed 2026-05-08 06:51 UTC · model grok-4.3

classification 📡 eess.SP

keywords DoA estimationmodel order selectioninformation theoretic criteriaorthogonal least squarestarget number estimationarray signal processingAICBIC

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The pith

Integrating information theoretic criteria into orthogonal least squares enables joint estimation of the number of targets and their directions of arrival.

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

The paper develops methods to combine information theoretic criteria for selecting the number of targets directly into the orthogonal least squares procedure for estimating their directions of arrival from antenna array data. Direct application of these criteria requires expensive maximum-likelihood searches over all angle combinations for each possible target count. Three integration approaches are derived: disjoint rank-based, joint selection-based, and hybrid rank-and-selection-based, each under both Akaike and Bayesian information criteria. Simulations show the hybrid version outperforms the other two variants and a literature baseline.

Core claim

The hybrid rank-and-selection-based ITC-OLS algorithm embeds the information theoretic penalty inside the greedy orthogonal least squares steps to jointly determine the model order and the DoA estimates, and numerical simulations establish that this hybrid consistently outperforms the disjoint and joint variants as well as a baseline method.

What carries the argument

The hybrid rank-and-selection-based ITC-OLS algorithm, which folds the complexity penalty into each greedy step of orthogonal least squares to decide both when to stop adding targets and which directions to select.

If this is right

The approach lowers complexity by avoiding full multidimensional searches for each candidate model order.
The same framework works for both Akaike and Bayesian information criteria.
It delivers better target count accuracy and DoA estimates than the compared methods in simulations.
The method applies directly to radar and array-based localization tasks requiring simultaneous detection and angle estimation.

Where Pith is reading between the lines

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

Similar penalty integration could be tested in other greedy sparse recovery algorithms for unknown source counts.
Validation on measured array data rather than only simulations would check performance under real propagation conditions.
The technique may connect to compressive sensing problems where both support size and parameter values must be recovered.

Load-bearing premise

Embedding the information theoretic penalty inside the sequential greedy steps of orthogonal least squares preserves the statistical consistency of model-order selection.

What would settle it

Array data with a known number of targets in which the hybrid ITC-OLS selects the wrong count more often than an exhaustive maximum-likelihood ITC method, especially for closely spaced targets or low signal-to-noise ratios.

Figures

Figures reproduced from arXiv: 2605.06198 by Fran\c{c}ois Horlin, Gilles Monnoyer, J\'er\^ome Louveaux, Martin Willame.

**Figure 1.** Figure 1: Comparison of the performance of the different ITC-OLS algorithms in terms of the Youden’s view at source ↗

read the original abstract

We address the joint estimation of the number of targets and their direction-of-arrivals (DoAs) using antenna arrays. Target-number estimation can be formulated as a model-order selection problem and solved with the information theoretic criteria (ITC). The ITC minimize an objective function that balances a likelihood term and a complexity penalty. However, direct application of the ITC requires maximum-likelihood DoA estimates for each candidate model order, which is computationally prohibitive because it entails a multidimensional search over all angle combinations. To reduce complexity, many radar processing exploit greedy methods such as orthogonal least squares (OLS). In this paper, we explore three distinct methods to integrate the ITC model-order selection into the OLS estimation procedure for joint target-number and DoA estimation. Specifically, we propose the disjoint rank-based, the joint selection-based, and the hybrid rank-and-selection-based ITC-OLS algorithms. Each algorithm is derived under both the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) frameworks. Numerical simulations show that the proposed hybrid ITC-OLS algorithm consistently outperforms both the other proposed variants and a baseline method from the literature.

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.

Desk Editor's Note private letter to a colleague

The paper gives three concrete recipes for folding AIC/BIC into OLS so you can jointly pick the number of targets and their angles without a full multi-dimensional ML search each time.

read the letter

The hybrid ITC-OLS variant is the one worth looking at first. It combines rank information with selection penalties inside the greedy steps and, according to the simulations, beats both the disjoint and joint variants plus a literature baseline. That is the practical payoff: you keep the low complexity of orthogonal least squares while still getting a model-order decision out of the same run. The three integration strategies are spelled out clearly enough that someone could implement them from the description. The simulations are the main evidence offered, and they show consistent outperformance for the hybrid under the tested conditions. That is useful for anyone who needs faster joint estimation in radar or array processing and is willing to validate on their own scenarios. The soft spot is exactly the one the stress-test flags. Because OLS builds the support sequentially by residual reduction, the ITC penalty is only evaluated along that particular path rather than over all subsets of a given size. Nothing in the work bounds how much this path dependence can shift the selected order away from the global ITC minimum, especially when sources are close or SNR is moderate. There is also no analytic consistency argument, only Monte Carlo results whose scenario coverage and controls are not detailed in the abstract. The paper is aimed at engineers who need implementable algorithms more than at theorists who want guarantees. A reader already working with greedy DoA methods would find the variants and the hybrid comparison worth trying. I would send it to peer review. The algorithmic contribution is clear and the simulation evidence is enough to justify referee time, even if the theoretical side stays light.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes three strategies for integrating information-theoretic criteria (AIC and BIC) into the orthogonal least squares (OLS) greedy procedure to jointly estimate the number of targets and their directions-of-arrival (DoAs) from array observations. The strategies are a disjoint rank-based ITC-OLS, a joint selection-based ITC-OLS, and a hybrid rank-and-selection-based ITC-OLS. Numerical simulations are presented to show that the hybrid variant consistently outperforms the other two proposed variants as well as a literature baseline.

Significance. If the performance advantage holds under broader conditions, the hybrid integration offers a practical, lower-complexity route to model-order selection in DoA estimation that avoids exhaustive multidimensional ML searches. The explicit comparison of the three integration styles supplies useful guidance on the trade-off between rank information and selection penalties in greedy array processing.

major comments (2)

[Hybrid ITC-OLS description (likely §3.3)] Hybrid ITC-OLS description (likely §3.3): embedding the ITC penalty inside the sequential OLS atom-selection steps evaluates the criterion only along one greedy residual path rather than over all subsets of a given cardinality. This path dependence can produce model-order errors that a global combinatorial ITC minimization would avoid, especially for closely spaced sources or moderate SNR; no consistency analysis or counter-example study is supplied to quantify the resulting bias.
[Numerical results section (likely §4)] Numerical results section (likely §4): the reported outperformance of the hybrid variant rests on Monte-Carlo trials whose exact parameters (number of trials, array size and geometry, source correlation, angular separation grid, SNR range, and any statistical significance testing of the differences) are not fully specified. Without these controls it is difficult to judge whether the superiority is robust or scenario-specific.

minor comments (2)

[Abstract and Introduction] The abstract and introduction could explicitly quantify the complexity saving relative to direct multi-dimensional ML-ITC (e.g., big-O scaling with number of targets and grid size).
[Algorithm derivations] Notation for the residual norm and the ITC penalty term should be unified across the three algorithm descriptions to avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the hybrid ITC-OLS algorithm and the simulation setup. We address each major comment below, acknowledging valid points where the manuscript can be strengthened through clarification or added discussion.

read point-by-point responses

Referee: Hybrid ITC-OLS description (likely §3.3): embedding the ITC penalty inside the sequential OLS atom-selection steps evaluates the criterion only along one greedy residual path rather than over all subsets of a given cardinality. This path dependence can produce model-order errors that a global combinatorial ITC minimization would avoid, especially for closely spaced sources or moderate SNR; no consistency analysis or counter-example study is supplied to quantify the resulting bias.

Authors: We agree that the hybrid ITC-OLS remains a greedy procedure that follows a single residual path and therefore constitutes an approximation rather than an exhaustive combinatorial minimization of the ITC. The hybrid variant was specifically designed to combine residual-rank information with the ITC penalty at each selection step, which our simulations indicate improves model-order accuracy relative to the purely rank-based and purely selection-based variants. A full asymptotic consistency analysis lies outside the scope of the present work, which emphasizes practical low-complexity algorithms. We will revise Section 3.3 to explicitly note the path-dependent nature of the approach and its possible limitations for closely spaced sources or moderate SNR, and we will add a short illustrative simulation example in Section 4 showing a scenario where the hybrid method selects an incorrect order. revision: partial
Referee: Numerical results section (likely §4): the reported outperformance of the hybrid variant rests on Monte-Carlo trials whose exact parameters (number of trials, array size and geometry, source correlation, angular separation grid, SNR range, and any statistical significance testing of the differences) are not fully specified. Without these controls it is difficult to judge whether the superiority is robust or scenario-specific.

Authors: The simulation parameters are already stated in the opening paragraph of Section 4, but we accept that they are not presented in a consolidated, easily verifiable form. In the revised manuscript we will insert a new Table I that explicitly tabulates all experimental settings: 5000 Monte-Carlo trials, an 8-element uniform linear array with half-wavelength spacing, uncorrelated sources, angular separations ranging from 5° to 30° in 5° increments, SNR values from -10 dB to 20 dB, and the absence of formal hypothesis testing (performance differences are reported via mean success rates with standard-deviation shading in the figures). These additions will allow readers to reproduce the exact conditions under which the hybrid variant outperforms the baselines. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic proposal evaluated on independent Monte-Carlo trials

full rationale

The paper proposes three algorithmic variants (disjoint rank-based, joint selection-based, hybrid rank-and-selection ITC-OLS) that embed AIC/BIC penalties inside the OLS greedy procedure for joint model-order and DoA estimation. No derivation chain exists that reduces a claimed result to its own inputs by construction: there are no fitted parameters renamed as predictions, no self-definitional equations, and no load-bearing self-citations or uniqueness theorems invoked. The central claim of consistent outperformance rests on numerical simulations across independent Monte-Carlo trials rather than analytic identities or self-referential fits. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard linear-array signal models and Gaussian noise assumptions already present in the DoA literature; no new free parameters, axioms, or invented entities are introduced beyond the algorithmic combinations themselves.

axioms (1)

domain assumption Linear array signal model with additive white Gaussian noise.
Required for the validity of OLS and for the likelihood term inside AIC/BIC.

pith-pipeline@v0.9.0 · 5514 in / 1142 out tokens · 58139 ms · 2026-05-08T06:51:18.642335+00:00 · methodology

discussion (0)

Reference graph

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