Rank-Sparsity Incoherence for Matrix Decomposition

Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification, and is NP-hard in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components, by minimizing a linear combination of the $\ell_1$ norm and the nuclear norm of the components. We develop a notion of \emph{rank-sparsity incoherence}, expressed as an uncertainty principle between the sparsity pattern of a matrix and its row and column spaces, and use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature, with the tangent spaces to the algebraic varieties of sparse and low-rank matrices playing a prominent role. When the sparse and low-rank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems

Coherency Matrix Decomposition-Based Polarimetric Persistent Scatterer Interferometry

© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.The rationale of polarimetric optimization techniques is to enhance the phase quality of the interferograms by combining adequately the different polarization channels available to produce an improved one. Different approaches have been proposed for polarimetric persistent scatterer interferometry (PolPSI). They range from the simple and computationally efficient BEST, where, for each pixel, the polarimetric channel with the best response in terms of phase quality is selected, to those with high-computational burden like the equal scattering mechanism (ESM) and the suboptimum scattering mechanism (SOM). BEST is fast and simple, but it does not fully exploit the potentials of polarimetry. On the other side, ESM explores all the space of solutions and finds the optimal one but with a very high-computational burden. A new PolPSI algorithm, named coherency matrix decomposition-based PolPSI (CMD-PolPSI), is proposed to achieve a compromise between phase optimization and computational cost. Its core idea is utilizing the polarimetric synthetic aperture radar (PolSAR) coherency matrix decomposition to determine the optimal polarization channel for each pixel. Three different PolSAR image sets of both full- (Barcelona) and dual-polarization (Murcia and Mexico City) are used to evaluate the performance of CMD-PolPSI. The results show that CMD-PolPSI presents better optimization results than the BEST method by using either $D_{\mathrm{ A}}$ or temporal mean coherence as phase quality metrics. Compared with the ESM algorithm, CMD-PolPSI is 255 times faster but its performance is not optimal. The influence of the number of available polarization channels and pixel's resolutions on the CMD-PolPSI performance is also discussed.Peer ReviewedPostprint (author's final draft

High Dimensional Low Rank plus Sparse Matrix Decomposition

This paper is concerned with the problem of low rank plus sparse matrix decomposition for big data. Conventional algorithms for matrix decomposition use the entire data to extract the low-rank and sparse components, and are based on optimization problems with complexity that scales with the dimension of the data, which limits their scalability. Furthermore, existing randomized approaches mostly rely on uniform random sampling, which is quite inefficient for many real world data matrices that exhibit additional structures (e.g. clustering). In this paper, a scalable subspace-pursuit approach that transforms the decomposition problem to a subspace learning problem is proposed. The decomposition is carried out using a small data sketch formed from sampled columns/rows. Even when the data is sampled uniformly at random, it is shown that the sufficient number of sampled columns/rows is roughly O(r\mu), where \mu is the coherency parameter and r the rank of the low rank component. In addition, adaptive sampling algorithms are proposed to address the problem of column/row sampling from structured data. We provide an analysis of the proposed method with adaptive sampling and show that adaptive sampling makes the required number of sampled columns/rows invariant to the distribution of the data. The proposed approach is amenable to online implementation and an online scheme is proposed.Comment: IEEE Transactions on Signal Processin