arXiv subjects

Levent Tunçel

Publications and source records attributed to Levent Tunçel.

38 records · Page 3Linked to original sources

arXiv2011

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 $\N¶$-hard and a popular approach replaces the rank function with the nuclear norm of the matrix variable. In this paper, we extend the concept of $s$-goodness for a sensing matrix in sparse signal recovery (proposed by Juditsky and Nemirovski [Math Program, 2011]) to linear transformations in LMR. Then, we give characterizations of $s$-goodness in the context of LMR. Using the two characteristic $s$-goodness constants, $γ_s$ and $\hatγ_s$, of a linear transformation, not only do we derive necessary and sufficient conditions for a linear transformation to be $s$-good, but also provide sufficient conditions for exact and stable $s$-rank matrix recovery via the nuclear norm minimization under mild assumptions. Moreover, we give computable upper bounds for one of the $s$-goodness characteristics which leads to verifiable sufficient conditions for exact low-rank matrix recovery.

cs.IT↗

arXiv2010

Homogeneous Cone Complementarity Problems and $P$ 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 order-$P_0$ and $R_0$, or the $P_0$ and $R_0$ properties then all the associated HCCPs have solutions. In particular, if a continuous function has the trace-$P$ property then the associated HCCP has a unique solution (if any); if it has the uniform-trace-$P$ property then the associated HCCP has the global uniqueness (of the solution) property (GUS). We present a necessary condition for a nonlinear transformation to have the GUS property. Moreover, we establish a global error bound for the HCCP with the uniform-trace-$P$ property. Finally, we study the HCCP with the relaxation transformation on a $T$-algebra and automorphism invariant properties for homogeneous cone linear complementarity problem.

math.OC↗