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The exact group-sparse recovery for block diagonal matrices with subexponential entries

Guozheng Dai, Tiankun Diao, Hanchao Wang

Abstract

We study block-diagonal random matrices with i.i.d. subexponential entries and show that, despite their highly structured form, they already guarantee exact sparse recovery from a nearly optimal number of measurements. When the matrix reduces to a single block, our framework collapses to the classical i.i.d. subexponential ensemble, and our bounds recover the well-known optimal rates previously established for unstructured random matrices.

The exact group-sparse recovery for block diagonal matrices with subexponential entries

Abstract

We study block-diagonal random matrices with i.i.d. subexponential entries and show that, despite their highly structured form, they already guarantee exact sparse recovery from a nearly optimal number of measurements. When the matrix reduces to a single block, our framework collapses to the classical i.i.d. subexponential ensemble, and our bounds recover the well-known optimal rates previously established for unstructured random matrices.

Paper Structure

This paper contains 11 sections, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. $\alpha$-subexponential random variables
  5. $\varepsilon$-net
  6. Restricted isometry property
  7. Block diagonal matrices with subexponential entries
  8. The concentration inequality of $\Vert\Phi\Psi x\Vert_{1}$
  9. The uniform upper bound
  10. The uniform lower bound
  11. Proof of the main result

Theorems & Definitions (7)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['Theo_matrix_satisfies_rip']}
  • proof : Proof of Theorem \ref{['Theo_main']}