# Block-sparse Recovery of Semidefinite Systems and Generalized Null Space Conditions

## TL;DR

This paper introduces null space properties for recovering low-rank block-diagonal matrices via convex nuclear-norm minimization, providing new theoretical conditions and a deterministic class satisfying these properties.

## Contribution

It develops a general framework for null space conditions in block-sparse semidefinite systems, unifying and extending existing NSPs.

## Key findings

- Derived new null space properties for block-diagonal matrices.
- Established a deterministic class satisfying the NSP.
- Unified existing NSPs within a general setup.

## Abstract

This article considers the recovery of low-rank matrices via a convex nuclear-norm minimization problem and presents two null space properties (NSP) which characterize uniform recovery for the case of block-diagonal matrices and block-diagonal positive semidefinite matrices. These null-space conditions turn out to be special cases of a new general setup, which allows to derive the mentioned NSPs and well-known NSPs from the literature. We discuss the relative strength of these conditions and also present a deterministic class of matrices that satisfies the block-diagonal semidefinite NSP.