On Verifiable Sufficient Conditions for Sparse Signal Recovery via ℓ1\ell_1 Minimization

We propose novel necessary and sufficient conditions for a sensing matrix to be "$s$-good" - to allow for exact $\ell_1$-recovery of sparse signals with $s$ nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect $\ell_1$-recovery (nonzero measurement noise, nearly $s$-sparse signal, near-optimal solution of the optimization problem yielding the $\ell_1$-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse $\ell_1$-recovery and to efficiently computable upper bounds on those $s$ for which a given sensing matrix is $s$-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like Restricted Isometry or Restricted Eigenvalue properties

Joint Sensing Matrix and Sparsifying Dictionary Optimization for Tensor Compressive Sensing.

Tensor compressive sensing (TCS) is a multidimensional framework of compressive sensing (CS), and it is advantageous in terms of reducing the amount of storage, easing hardware implementations, and preserving multidimensional structures of signals in comparison to a conventional CS system. In a TCS system, instead of using a random sensing matrix and a predefined dictionary, the average-case performance can be further improved by employing an optimized multidimensional sensing matrix and a learned multilinear sparsifying dictionary. In this paper, we propose an approach that jointly optimizes the sensing matrix and dictionary for a TCS system. For the sensing matrix design in TCS, an extended separable approach with a closed form solution and a novel iterative nonseparable method are proposed when the multilinear dictionary is fixed. In addition, a multidimensional dictionary learning method that takes advantages of the multidimensional structure is derived, and the influence of sensing matrices is taken into account in the learning process. A joint optimization is achieved via alternately iterating the optimization of the sensing matrix and dictionary. Numerical experiments using both synthetic data and real images demonstrate the superiority of the proposed approache

Deriving RIP sensing matrices for sparsifying dictionaries

Compressive sensing involves the inversion of a mapping $SD \in \mathbb{R}^{m \times n}$, where $m < n$, $S$ is a sensing matrix, and $D$ is a sparisfying dictionary. The restricted isometry property is a powerful sufficient condition for the inversion that guarantees the recovery of high-dimensional sparse vectors from their low-dimensional embedding into a Euclidean space via convex optimization. However, determining whether $SD$ has the restricted isometry property for a given sparisfying dictionary is an NP-hard problem, hampering the application of compressive sensing. This paper provides a novel approach to resolving this problem. We demonstrate that it is possible to derive a sensing matrix for any sparsifying dictionary with a high probability of retaining the restricted isometry property. In numerical experiments with sensing matrices for K-SVD, Parseval K-SVD, and wavelets, our recovery performance was comparable to that of benchmarks obtained using Gaussian and Bernoulli random sensing matrices for sparse vectors

Recovery of Low-Rank Plus Compressed Sparse Matrices with Application to Unveiling Traffic Anomalies

Given the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish deterministic conditions under which exact recovery of the low-rank and sparse components becomes possible. This fundamental identifiability issue arises with traffic anomaly detection in backbone networks, and subsumes compressed sensing as well as the timely low-rank plus sparse matrix recovery tasks encountered in matrix decomposition problems. Leveraging the ability of $\ell_1$- and nuclear norms to recover sparse and low-rank matrices, a convex program is formulated to estimate the unknowns. Analysis and simulations confirm that the said convex program can recover the unknowns for sufficiently low-rank and sparse enough components, along with a compression matrix possessing an isometry property when restricted to operate on sparse vectors. When the low-rank, sparse, and compression matrices are drawn from certain random ensembles, it is established that exact recovery is possible with high probability. First-order algorithms are developed to solve the nonsmooth convex optimization problem with provable iteration complexity guarantees. Insightful tests with synthetic and real network data corroborate the effectiveness of the novel approach in unveiling traffic anomalies across flows and time, and its ability to outperform existing alternatives.Comment: 38 pages, submitted to the IEEE Transactions on Information Theor

CSWA: Aggregation-Free Spatial-Temporal Community Sensing

In this paper, we present a novel community sensing paradigm -- {C}ommunity {S}ensing {W}ithout {A}ggregation}. CSWA is designed to obtain the environment information (e.g., air pollution or temperature) in each subarea of the target area, without aggregating sensor and location data collected by community members. CSWA operates on top of a secured peer-to-peer network over the community members and proposes a novel \emph{Decentralized Spatial-Temporal Compressive Sensing} framework based on \emph{Parallelized Stochastic Gradient Descent}. Through learning the \emph{low-rank structure} via distributed optimization, CSWA approximates the value of the sensor data in each subarea (both covered and uncovered) for each sensing cycle using the sensor data locally stored in each member's mobile device. Simulation experiments based on real-world datasets demonstrate that CSWA exhibits low approximation error (i.e., less than $0.2 ^\circ$C in city-wide temperature sensing task and $10$ units of PM2.5 index in urban air pollution sensing) and performs comparably to (sometimes better than) state-of-the-art algorithms based on the data aggregation and centralized computation.Comment: This paper has been accepted by AAAI 2018. First two authors are equally contribute

Model-based Optimization of Compressive Antennas for High-Sensing-Capacity Applications

This paper presents a novel, model-based compressive antenna design method for high sensing capacity imaging applications. Given a set of design constraints, the method maximizes the sensing capacity of the compressive antenna by varying the constitutive properties of scatterers distributed along the antenna. Preliminary 2D design results demonstrate the new method's ability to produce antenna configurations with enhanced imaging capabilities