arXiv subjects

Maxime Ferreira Da Costa

Publications and source records attributed to Maxime Ferreira Da Costa.

At least 19 recordsLinked to original sources

arXiv2026

Characterization of the Basin of Convexity for Multi-Snapshot Spike Deconvolution via Variable Projection

The problem of multi-snapshot spike deconvolution is studied, where the goal is to recover the locations of sparse impulses from their noisy convolution with a known point spread function (PSF) across multiple snapshots. A variable-projection formulation is adopted, in which the amplitudes are eliminated in closed form, thereby reducing the task to a nonconvex least-squares problem over the spike locations alone. This formulation is referred to as the variable-projection formulation of spike deconvolution (VarProSD). An explicit characterization of the basin of convexity of the VarProSD objective is provided in terms of key PSF properties, including its power spectral density and smoothness, revealing how sampling bandwidth and spike separation affect the local geometry. Within this basin, consistency of the estimator in the number of snapshots is established under stochastic noise, and a complementary, sharper error bound is derived under adversarial noise through the local Lipschitz property of the inverse map. Local convergence guarantees for gradient descent are further established when initialization is performed within the basin. A central role throughout the analysis is played by Beurling--Selberg extremal approximations, which enable sharp, PSF-agnostic bounds on the conditioning of the structured matrices arising in the optimization landscape. Numerical experiments are presented to corroborate the theoretical findings and demonstrate the effectiveness of modified ESPRIT initialization followed by gradient-based refinement.

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arXiv2026

Atomic Hybrid Sparse/Diffuse Channel Estimation and Cramér-Rao Bounds Analysis

In this paper, an atomic hybrid sparse/diffuse (aHSD) channel model in the frequency domain is proposed. Based on a structural analysis of the resolvable paths and diffuse scattering statistics, the Hybrid Atomic-Least-Squares (HALS) algorithm is designed to estimate sparse/diffuse components with a combined atomic and $\ell_2$ regularization. A theoretical analysis of the Lagrange dual problem is conducted, and the conditions required for primal and dual solutions are provided, supporting an off-the-grid delay-time estimator. The Cramér--Rao Bound (CRB) analysis in this paper focuses on the estimation of the channel parameters, resulting in a bound on the aggregate channel. Lower and upper bounds for the CRB on parameters are derived as functions of the minimum separations between frequency parameters. Numerical results via simulations on synthetic and real data validate the efficacy of the HALS estimation strategy and show the improved predictive ability of the CRB analysis for the performance of HALS versus previously considered bounds.

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arXiv2026

Local Geometry of Least Squares for Unmixing Signals with Parameter-Dependent Dictionaries

Modeling signals as linear combinations of atoms from a dictionary is ubiquitous in modern signal processing. In the finite-dimensional setting, whenever atoms depend nonlinearly upon unknown parameters, the signal model is said to be separable. In this work, we study least-squares reconstruction of separable signals and establish a unified theoretical framework for their analysis. We introduce the unmixing metric, a distance that captures the distinct roles and sensitivities of linear and nonlinear parameters, and establish local convergence and stability guarantees under its topology. We then analyze variable projection from a geometric perspective, showing that it corresponds to restricting the optimization to the manifold of optimal linear parameters. This viewpoint provides a principled explanation for the improved algorithmic behavior of variable projection observed in practice, and produces sharp theoretical guarantees. The generic theory for separable problems is specialized to the case of point spread function (PSF) unmixing. We introduce a parametric notion of coherence and show that support separation directly controls both the size of the convergence region and the stability of recovery. Numerical experiments corroborate the theoretical predictions and demonstrate the practical relevance of the proposed framework.

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arXiv2026

Goal-oriented Resource Allocation for Collaborative Integrated Sensing and Communication

In this paper, we consider resource allocation for a collaborative integrated sensing and communication (ISAC) scenario, in which distributed smart devices can be scheduled to perform sensing and transmit their sensing features to a fusion center. The fusion center aims to perform classification tasks on the environment based on received features. A scalable networksensing framework is proposed to balance the performance of the sensing service with that of the classical enhanced Mobile Broadband (eMBB) service. We adopt a tractable theoretical metric, the discriminant gain, as a proxy for the classification goal. We formulate cross-layer optimization problems to maximize discriminant gain under constraints on energy consumption and eMBB communication quality for the independent and joint scheduling policies. The joint scheduling policy has considerably higher complexity than the independent scheduling policy, in exchange for better collaborative sensing performance. A simplified gain model is proposed to reduce the complexity and practicality of the joint scheduling policy. Both policies are obtained via successive convex approximation and parametric convex optimization. Extensive experiments are conducted to verify the goal-oriented framework and the two policies. It is demonstrated that the two policies outperform the baseline policies with both synthetic and realistic radar simulation datasets. The joint scheduling policy can exploit device correlations and thus performs better than the independent scheduling policy under strong correlations and strict communication constraints.

eess.SY↗

arXiv2025

Strong Basin of Attraction for Unmixing Kernels With the Variable Projection Method

The problem of recovering a mixture of spike signals convolved with distinct point spread functions (PSFs) lying on a parametric manifold, under the assumption that the spike locations are known, is studied. The PSF unmixing problem is formulated as a projected non-linear least squares estimator. A lower bound on the radius of the region of strong convexity is established in the presence of noise as a function of the manifold coherence and Lipschitz properties, guaranteeing convergence and stability of the optimization program. Numerical experiments highlight the speed of decay of the PSF class in the problem's conditioning and confirm theoretical findings. Finally, the proposed estimator is deployed on real-world spectroscopic data from laser-induced breakdown spectroscopy (LIBS), removing the need for manual calibration and validating the method's practical relevance.

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arXiv2025

Convergence Guarantees for Unmixing PSFs over a Manifold with Non-Convex Optimization

The problem of recovering the parameters of a mixture of spike signals convolved with different PSFs is considered. Herein, the spike support is assumed to be known, while the PSFs lie on a manifold. A non-linear least squares estimator of the mixture parameters is formulated. In the absence of noise, a lower bound on the radius of the strong basin of attraction i.e., the region of convergence, is derived. Key to the analysis is the introduction of coherence and interference functions, which capture the conditioning of the PSF manifold in terms of the minimal separation of the support. Numerical experiments validate the theoretical findings. Finally, the practicality and efficacy of the non-linear least squares approach are showcased on spectral data from laser-induced breakdown spectroscopy.

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arXiv2025

Global Convergence of ESPRIT with Preconditioned First-Order Methods for Spike Deconvolution

Spike deconvolution is the problem of recovering point sources from their convolution with a known point spread function, playing a fundamental role in many sensing and imaging applications. This paper proposes a novel approach combining ESPRIT with Preconditioned Gradient Descent (PGD) to estimate the amplitudes and locations of the point sources by a non-linear least squares. The preconditioning matrices are adaptively designed to account for variations in the learning process, ensuring a proven super-linear convergence rate. We provide local convergence guarantees for PGD and performance analysis of ESPRIT reconstruction, leading to global convergence guarantees for our method in one-dimensional settings with multiple snapshots, demonstrating its robustness and effectiveness. Numerical simulations corroborate the performance of the proposed approach for spike deconvolution.

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arXiv2025

Physical Layer Location Privacy in SIMO Communication Using Fake Path Injection

Fake path injection is an emerging paradigm for inducing privacy over wireless networks. In this paper, fake paths are injected by the transmitters into a single-input multiple-output (SIMO) communication channel to obscure their physical location from an eavesdropper. The case where the receiver (Bob) and the eavesdropper (Eve) use a linear uniform array to locate the transmitter's (Alice) position is considered. A novel statistical privacy metric is defined as the ratio between the smallest (resp. largest) eigenvalues of Eve's (resp. Bob's) Cramér-Rao lower bound (CRB) on the SIMO channel parameters to assess the privacy enhancements. Leveraging the spectral properties of generalized Vandermonde matrices, bounds on the privacy margin of the proposed scheme are derived. Specifically, it is shown that the privacy margin increases quadratically in the inverse of the angular separation between the true and the fake paths under Eve's perspective. Numerical simulations validate the theoretical findings on CRBs and showcase the approach's benefit in terms of bit error rates achievable by Bob and Eve.

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arXiv2024

Guaranteed Private Communication with Secret Block Structure

A novel private communication framework is proposed where privacy is induced by transmitting over a channel instances of linear inverse problems that are identifiable to the legitimate receiver but unidentifiable to an eavesdropper. The gap in identifiability is created in the framework by leveraging secret knowledge between the transmitter and the legitimate receiver. Specifically, the case where the legitimate receiver harnesses a secret block structure to decode a transmitted block-sparse message from underdetermined linear measurements in conditions where classical compressed sensing would provably fail is examined. The applicability of the proposed scheme to practical multiple-access wireless communication systems is discussed. The protocol's privacy is studied under a single transmission, and under multiple transmissions without refreshing the secret block structure. It is shown that, under a specific scaling of the channel dimensions and transmission parameters, the eavesdropper can attempt to overhear the block structure from the fourth-order moments of the channel output. Computation of a statistical lower bound suggests that the proposed fourth-order moment secret block estimation strategy is near optimal. The performance of a spectral clustering algorithm is studied to that end, defining scaling laws on the lifespan of the secret key before the communication is compromised. Finally, numerical experiments corroborating the theoretical findings are conducted.

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arXiv2024

Small-Noise Sensitivity Analysis of Locating Pulses in the Presence of Adversarial Perturbation

A fundamental small-noise sensitivity analysis of spike localization in the presence of adversarial perturbations and an arbitrary point spread function (PSF) is presented. The analysis leverages the local Lipschitz property of the inverse map from measurement noise to parameter estimate. In the small noise regime, the local Lipschitz constant converges to the spectral norm of the noiseless Jacobian of the inverse map. An interpretable upper bound in terms of the minimum separation of spikes, norms, and flatness of the PSF and its derivative, as well as the distribution of spike amplitudes is provided. Numerical experiments highlighting the relevance of the theoretical bound as a proxy to the local Lipschitz constant and its dependence on the key attributes of the problem are presented.

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arXiv2023

Local Geometry of Nonconvex Spike Deconvolution from Low-Pass Measurements

Spike deconvolution is the problem of recovering the point sources from their convolution with a known point spread function, which plays a fundamental role in many sensing and imaging applications. In this paper, we investigate the local geometry of recovering the parameters of point sources$\unicode{x2014}$including both amplitudes and locations$\unicode{x2014}$by minimizing a natural nonconvex least-squares loss function measuring the observation residuals. We propose preconditioned variants of gradient descent (GD), where the search direction is scaled via some carefully designed preconditioning matrices. We begin with a simple fixed preconditioner design, which adjusts the learning rates of the locations at a different scale from those of the amplitudes, and show it achieves a linear rate of convergence$\unicode{x2014}$in terms of entrywise errors$\unicode{x2014}$when initialized close to the ground truth, as long as the separation between the true spikes is sufficiently large. However, the convergence rate slows down significantly when the dynamic range of the source amplitudes is large. To bridge this issue, we introduce an adaptive preconditioner design, which compensates for the learning rates of different sources in an iteration-varying manner based on the current estimate. The adaptive design provably leads to an accelerated convergence rate that is independent of the dynamic range, highlighting the benefit of adaptive preconditioning in nonconvex spike deconvolution. Numerical experiments are provided to corroborate the theoretical findings.

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arXiv2022

On the Stability of Super-Resolution and a Beurling-Selberg Type Extremal Problem

Super-resolution estimation is the problem of recovering a stream of spikes (point sources) from the noisy observation of a few numbers of its first trigonometric moments. The performance of super-resolution is recognized to be intimately related to the separation between the spikes to recover. A novel notion of stability of the Fisher information matrix (FIM) of the super-resolution problem is introduced when the minimal eigenvalue of the FIM is not asymptotically vanishing. The regime where the minimal separation is inversely proportional to the number of acquired moments is considered. It is shown that there is a separation threshold above which the eigenvalues of the FIM can be bounded by a quantity that does not depend on the number of moments. The proof relies on characterizing the connection between the stability of the FIM and a generalization of the Beurling-Selberg box approximation problem.

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arXiv2022

Joint Localization and Orientation Estimation in Millimeter-Wave MIMO OFDM Systems via Atomic Norm Minimization

Herein, an atomic norm based method for accurately estimating the location and orientation of a target from millimeter-wave multi-input-multi-output (MIMO) orthogonal frequency-division multiplexing (OFDM) signals is presented. A novel virtual channel matrix is introduced and an algorithm to extract localization-relevant channel parameters from its atomic norm decomposition is designed. Then, based on the extended invariance principle, a weighted least squares problem is proposed to accurately recover the location and orientation using both line-of-sight and non-line-of-sight channel information. The conditions for the optimality and uniqueness of the estimate and theoretical guarantees for the estimation error are characterized for the noiseless and the noisy scenarios. Theoretical results are confirmed via simulation. Numerical results investigate the robustness of the proposed algorithm to incorrect model order selection or synchronization error, and highlight performance improvements over a prior method. The resultant performance nearly achieves the Cramer-Rao lower bound on the estimation error.

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arXiv2021

A Framework for Private Communication with Secret Block Structure

Harnessing a block-sparse prior to recover signals through underdetermined linear measurements has been extensively shown to allow exact recovery in conditions where classical compressed sensing would provably fail. We exploit this result to propose a novel private communication framework where the secrecy is achieved by transmitting instances of an unidentifiable compressed sensing problem over a public channel. The legitimate receiver can attempt to overcome this ill-posedness by leveraging secret knowledge of a block structure that was used to encode the transmitter's message. We study the privacy guarantees of this communication protocol to a single transmission, and to multiple transmissions without refreshing the shared secret. Additionally, we propose an algorithm for an eavesdropper to learn the block structure via the method of moments and highlight the privacy benefits of this framework through numerical experiments.

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arXiv2021

Atomic Norm Based Localization and Orientation Estimation for Millimeter-Wave MIMO OFDM Systems

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arXiv2020

Compressed Super-Resolution of Positive Sources

Atomic norm minimization is a convex optimization framework to recover point sources from a subset of their low-pass observations, or equivalently the underlying frequencies of a spectrally-sparse signal. When the amplitudes of the sources are positive, a positive atomic norm can be formulated, and exact recovery can be ensured without imposing a separation between the sources, as long as the number of observations is greater than the number of sources. However, the classic formulation of the atomic norm requires to solve a semidefinite program involving a linear matrix inequality of a size on the order of the signal dimension, which can be prohibitive. In this letter, we introduce a novel "compressed" semidefinite program, which involves a linear matrix inequality of a reduced dimension on the order of the number of sources. We guarantee the tightness of this program under certain conditions on the operator involved in the dimensionality reduction. Finally, we apply the proposed method to direction finding over sparse arrays based on second-order statistics and achieve significant computational savings.

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arXiv2020

On the Stable Resolution Limit of Total Variation Regularization for Spike Deconvolution

The stability of spike deconvolution, which aims at recovering point sources from their convolution with a point spread function (PSF), is known to be related to the separation between those sources. When the observations are noisy, it is critical to ensure support stability, where the deconvolution does not lead to spurious, or oppositely, missing estimates of the point sources. In this paper, we study the resolution limit of stably recovering the support of two closely located point sources using the Beurling-LASSO estimator, which is a convex optimization approach based on total variation regularization. We establish a sufficient separation criterion between the sources, depending only on the PSF, above which the Beurling-LASSO estimator is guaranteed to return a stable estimate of the point sources, with the same number of estimated elements as that of the ground truth. Our result highlights the impact of PSF on the resolution limit in the noisy setting, which was not evident in previous studies of the noiseless setting. Towards the end, we show that the same resolution limit applies to resolving two close-located sources in conjunction of other well-separated sources.

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arXiv2019

Harnessing Sparsity over the Continuum: Atomic Norm Minimization for Super Resolution

Convex optimization recently emerges as a compelling framework for performing super resolution, garnering significant attention from multiple communities spanning signal processing, applied mathematics, and optimization. This article offers a friendly exposition to atomic norm minimization as a canonical convex approach to solve super resolution problems. The mathematical foundations and performances guarantees of this approach are presented, and its application in super resolution image reconstruction for single-molecule fluorescence microscopy are highlighted.

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