  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 measuring 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 elaborated for analyzing
both noiseless and noisy recovery conditions. Specifically, we present the
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 the
signal-to-noise ratio bound, are established to reveal the computable
theoretical interpretability based on the newly defined MIP, which we
introduce. The flexibility of tensor recovery is highlighted, i.e., the
reliable recovery can be guaranteed by optimizing MIP of the measurement matrix
set. Analytical comparisons demonstrate that the theoretical results developed
are tighter and less restrictive than the existing ones (if any). Further
discussions provide tensor extensions for several classic greedy algorithms,
indicating that the sophisticated results derived are universal and applicable
to all these tensorized variants.
