Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices
It provides foundational theoretical guarantees for compressed sensing and matrix recovery, which are critical for signal processing and machine learning applications.
This paper tackles the problem of recovering sparse signals and low-rank matrices in compressed sensing and affine rank minimization by establishing sharp restricted isometry conditions for exact and stable recovery in noiseless and noisy cases, showing that conditions like δ_{tk}^A < √((t-1)/t) guarantee exact recovery for all k-sparse signals.
This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while leads to sharp results. It is shown that for any given constant $t\ge {4/3}$, in compressed sensing $δ_{tk}^A < \sqrt{(t-1)/t}$ guarantees the exact recovery of all $k$ sparse signals in the noiseless case through the constrained $\ell_1$ minimization, and similarly in affine rank minimization $δ_{tr}^\mathcal{M}< \sqrt{(t-1)/t}$ ensures the exact reconstruction of all matrices with rank at most $r$ in the noiseless case via the constrained nuclear norm minimization. Moreover, for any $ε>0$, $δ_{tk}^A<\sqrt{\frac{t-1}{t}}+ε$ is not sufficient to guarantee the exact recovery of all $k$-sparse signals for large $k$. Similar result also holds for matrix recovery. In addition, the conditions $δ_{tk}^A < \sqrt{(t-1)/t}$ and $δ_{tr}^\mathcal{M}< \sqrt{(t-1)/t}$ are also shown to be sufficient respectively for stable recovery of approximately sparse signals and low-rank matrices in the noisy case.