Block-Sparse Tensor Recovery

Abstract

This work explores the fundamental problem of the recoverability of a sparse tensor being reconstructed from its compressed embodiment. We present a generalized model of block-sparse tensor recovery as a theoretical foundation, where concepts involving a holistic mutual incoherence property (MIP) of the measurement matrix set are defined. A representative algorithm based on the orthogonal matching pursuit (OMP) framework, called tensor generalized block OMP (T-GBOMP), is applied to the theoretical framework for analyzing both noiseless and noisy recovery conditions. Specifically, we present an exact recovery condition (ERC) and sufficient conditions for establishing it with consideration of different degrees of restriction. Reliable reconstruction conditions, in terms of the residual convergence, the estimated error and a signal-to-noise ratio bound, are established to reveal the computable theoretical interpretability based on the newly defined MIP. The flexibility of tensor recovery is highlighted, i.e., the reliable recovery can be guaranteed by optimizing the MIP of the measurement matrix set. Analytical comparisons demonstrate that the theoretical results developed are tighter and less restrictive than existing ones (if any). Further discussions provide tensor extensions for several classic greedy algorithms, indicating that the results derived are universal and applicable to all these tensorized variants.