Sparse Recovery on Euclidean Jordan Algebras

We consider the sparse recovery problem on Euclidean Jordan algebra (SREJA), which includes sparse signal recovery and low-rank symmetric matrix recovery as special cases. We introduce the restricted isometry property, null space property (NSP), and $s$-goodness for linear transformations in $s$-sparse element recovery on Euclidean Jordan algebra (SREJA), all of which provide sufficient conditions for … Read more

S-semigoodness for Low-Rank Semidefinite Matrix Recovery

We extend and characterize the concept of $s$-semigoodness for a sensing matrix in sparse nonnegative recovery (proposed by Juditsky , Karzan and Nemirovski [Math Program, 2011]) to the linear transformations in low-rank semidefinite matrix recovery. We show that s-semigoodness is not only a necessary and sufficient condition for exact $s$-rank semidefinite matrix recovery by a … Read more

The Nonnegative $ Norm Minimization under Generalized hBcmatrix Measurement

In this paper, we consider the $l_0$ norm minimization problem with linear equation and nonnegativity constraints. By introducing the concept of generalized $Z$-matrix for a rectangular matrix, we show that this $l_0$ norm minimization with such a kind of measurement matrices and nonnegative observations can be exactly solved via the corresponding $l_p$ ($0

Sufficient Conditions for Low-rank Matrix Recovery,Translated from Sparse Signal Recovery

The low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, system identification and control. This class of optimization problems is $NP$-hard and a popular approach replaces the rank function with the nuclear norm of the … Read more

New Bounds for Restricted Isometry Constants in Low-rank Matrix Recovery

In this paper, we establish new bounds for restricted isometry constants (RIC) in low-rank matrix recovery. Let $\A$ be a linear transformation from $\R^{m \times n}$ into $\R^p$, and $r$ the rank of recovered matrix $X\in \R^{m \times n}$. Our main result is that if the condition on RIC satisfies $\delta_{2r+k}+2(\frac{r}{k})^{1/2}\delta_{\max\{r+\frac{3}{2}k,2k\}}

Homogeneous Cone Complementarity Problems and $ Properties

We consider existence and uniqueness properties of a solution to homogeneous cone complementarity problem (HCCP). Employing the $T$-algebraic characterization of homogeneous cones, we generalize the $P, P_0, R_0$ properties for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of HCCP. We prove that if a continuous function has either the … Read more

Fischer-Burmeister Complementarity Function on Euclidean Jordan Algebras

Recently, Gowda et al. [10] established the Fischer-Burmeister (FB) complementarity function (C-function) on Euclidean Jordan algebras. In this paper, we prove that FB C-function as well as the derivatives of the squared norm of FB C-function are Lipschitz continuous. Citation Research Report CORR 2007-17, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, … Read more

Monotonicity of L”{o}wner Operators and Its Applications to Symmetric Cone Complementarity Problems

This paper focuses on monotone L\”{o}wner operators in Euclidean Jordan algebras and their applications to the symmetric cone complementarity problem (SCCP). We prove necessary and sufficient conditions for locally Lipschitz L\”{o}wner operators to be monotone, strictly monotone and strongly monotone. We also study the relationship between monotonicity and operator-monotonicity of L\”{o}wner operators. As a by-product … Read more