arXiv subjects

Seyedahmad Mousavi

Publications and source records attributed to Seyedahmad Mousavi.

5 recordsLinked to original sources

arXiv2020

Exact Support and Vector Recovery of Constrained Sparse Vectors via Constrained Matching Pursuit

Matching pursuit, especially its orthogonal version (OMP) and variations, is a greedy algorithm widely used in signal processing, compressed sensing, and sparse modeling. Inspired by constrained sparse signal recovery, this paper proposes a constrained matching pursuit algorithm and develops conditions for exact support and vector recovery on constraint sets via this algorithm. We show that exact recovery via constrained matching pursuit not only depends on a measurement matrix but also critically relies on a constraint set. We thus identify an important class of constraint sets, called coordinate projection admissible set, or simply CP admissible sets; analytic and geometric properties of these sets are established. We study exact vector recovery on convex, CP admissible cones for a fixed support. We provide sufficient exact recovery conditions for a general support as well as necessary and sufficient recovery conditions when a support has small size. As a byproduct, we construct a nontrivial counterexample to a renowned necessary condition of exact recovery via the OMP for a support of size three. Moreover, using the properties of convex CP admissible sets and convex optimization techniques, we establish sufficient conditions for uniform exact recovery on convex CP admissible sets in terms of the restricted isometry-like constant and the restricted orthogonality-like constant.

math.OC↗

arXiv2019

Quantum Annealing Based Binary Compressive Sensing with Matrix Uncertainty

Compressive sensing is a novel approach that linearly samples sparse or compressible signals at a rate much below the Nyquist-Shannon sampling rate and outperforms traditional signal processing techniques in acquiring and reconstructing such signals. Compressive sensing with matrix uncertainty is an extension of the standard compressive sensing problem that appears in various applications including but not limited to cognitive radio sensing, calibration of the antenna, and deconvolution. The original problem of compressive sensing is NP-hard so the traditional techniques, such as convex and nonconvex relaxations and greedy algorithms, apply stringent constraints on the measurement matrix to indirectly handle this problem in the realm of classical computing. We propose well-posed approaches for both binary compressive sensing and binary compressive sensing with matrix uncertainty problems that are tractable by quantum annealers. Our approach formulates an Ising model whose ground state represents a sparse solution for the binary compressive sensing problem and then employs an alternating minimization scheme to tackle the binary compressive sensing with matrix uncertainty problem. This setting only requires the solution uniqueness of the considered problem to have a successful recovery process, and therefore the required conditions on the measurement matrix are notably looser. As a proof of concept, we can demonstrate the applicability of the proposed approach on the D-Wave quantum annealers; however, we can adapt our method to employ other modern computing phenomena -like adiabatic quantum computers (in general), CMOS annealers, optical parametric oscillators, and neuromorphic computing.

cs.IT↗

arXiv2018

The Hyper-Zagreb Index of Trees and Unicyclic Graphs

Applications in chemistry motivated mathematicians to define different topological indices for different types of graphs. The Hyper-Zagreb index (HM) is an important tool as it integrates the first and the second Zagreb indices. In this paper, we characterize the trees and unicyclic graphs with the first four and first eight greatest HM-value, respectively.

math.CO↗

arXiv2017

Solution Uniqueness of Convex Piecewise Affine Functions Based Optimization with Applications to Constrained $\ell_1$ Minimization

In this paper, we study the solution uniqueness of an individual feasible vector of a class of convex optimization problems involving convex piecewise affine functions and subject to general polyhedral constraints. This class of problems incorporates many important polyhedral constrained $\ell_1$ recovery problems arising from sparse optimization, such as basis pursuit, LASSO, and basis pursuit denoising, as well as polyhedral gauge recovery. By leveraging the max-formulation of convex piecewise affine functions and convex analysis tools, we develop dual variables based necessary and sufficient uniqueness conditions via simple and yet unifying approaches; these conditions are applied to a wide range of $\ell_1$ minimization problems under possible polyhedral constraints. An effective linear program based scheme is proposed to verify solution uniqueness conditions. The results obtained in this paper not only recover the known solution uniqueness conditions in the literature by removing restrictive assumptions but also yield new uniqueness conditions for much broader constrained $\ell_1$-minimization problems.

math.OC↗

arXiv2017

Least Sparsity of $p$-norm based Optimization Problems with $p > 1$

Motivated by $\ell_p$-optimization arising from sparse optimization, high dimensional data analytics and statistics, this paper studies sparse properties of a wide range of $p$-norm based optimization problems with $p > 1$, including generalized basis pursuit, basis pursuit denoising, ridge regression, and elastic net. It is well known that when $p > 1$, these optimization problems lead to less sparse solutions. However, the quantitative characterization of the adverse sparse properties is not available. In this paper, by exploiting optimization and matrix analysis techniques, we give a systematic treatment of a broad class of $p$-norm based optimization problems for a general $p > 1$ and show that optimal solutions to these problems attain full support, and thus have the least sparsity, for almost all measurement matrices and measurement vectors. Comparison to $\ell_p$-optimization with $0 < p \le 1$ and implications to robustness are also given. These results shed light on analysis and computation of general $p$-norm based optimization problems in various applications.

math.OC↗