arXiv subjects

Fatma Kilinc Karzan

Publications and source records attributed to Fatma Kilinc Karzan.

3 recordsLinked to original sources

arXiv2012

On unified view of nullspace-type conditions for recoveries associated with general sparsity structures

We discuss a general notion of "sparsity structure" and associated recoveries of a sparse signal from its linear image of reduced dimension possibly corrupted with noise. Our approach allows for unified treatment of (a) the "usual sparsity" and "usual $\ell_1$ recovery," (b) block-sparsity with possibly overlapping blocks and associated block-$\ell_1$ recovery, and (c) low-rank-oriented recovery by nuclear norm minimization. The proposed recovery routines are natural extensions of the usual $\ell_1$ minimization used in Compressed Sensing. Specifically we present nullspace-type sufficient conditions for the recovery to be precise on sparse signals in the noiseless case. Then we derive error bounds for imperfect (nearly sparse signal, presence of observation noise, etc.) recovery under these conditions. In all of these cases, we present efficiently verifiable sufficient conditions for the validity of the associated nullspace properties.

math.OC↗

arXiv2010

On Verifiable Sufficient Conditions for Sparse Signal Recovery via $\ell_1$ 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.

math.OC↗

arXiv2010

Verifiable conditions of $\ell_1$-recovery of sparse signals with sign restrictions

We propose necessary and sufficient conditions for a sensing matrix to be "s-semigood" -- to allow for exact $\ell_1$-recovery of sparse signals with at most $s$ nonzero entries under sign restrictions on part of the entries. We express the error bounds for imperfect $\ell_1$-recovery in terms of the characteristics underlying these conditions. Furthermore, we demonstrate 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 $s$-semigood. We concentrate on the properties of proposed verifiable sufficient conditions of $s$-semigoodness and describe their limits of performance.

math.ST↗