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Functional Donoho-Elad-Gribonval-Nielsen-Fuchs Sparsity Theorem

K. Mahesh Krishna

TL;DR

The paper extends the classical DEGN sparsity theorem from Hilbert spaces to abstract Banach spaces by employing 1-approximate Schauder frames (1-ASF). It introduces the analysis/synthesis framework in Banach spaces, defines a Banach-space Null Space Property (NSP), and proves that, under a bound involving $\sup_{n\neq m}|f_n(\tau_m)|$, a sparse coefficient $c$ with $\|c\|_0$ below this threshold yields a unique $\ell^1$-minimizer (problem $P_1$) and unique $\ell^0$-minimizer (problem $P_0$) for representations $x=\theta_\tau c$. This functional sparsity theorem thereby generalizes the DEGN result to a broader class of spaces, enabling sparse recovery in Banach-space settings and broadening potential applications in functional analysis and signal processing.

Abstract

Celebrated breakthrough sparsity theorem obtained independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]} and Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard $\ell_0$-minimization problem can be obtained using unique solution to P-Type $\ell_1$-minimization problem. In this paper, we extend their result to abstract Banach spaces using 1-approximate Schauder frames. We notice that the `normalized' condition for Hilbert spaces can be generalized to a larger extent when we consider Banach spaces.

Functional Donoho-Elad-Gribonval-Nielsen-Fuchs Sparsity Theorem

TL;DR

The paper extends the classical DEGN sparsity theorem from Hilbert spaces to abstract Banach spaces by employing 1-approximate Schauder frames (1-ASF). It introduces the analysis/synthesis framework in Banach spaces, defines a Banach-space Null Space Property (NSP), and proves that, under a bound involving , a sparse coefficient with below this threshold yields a unique -minimizer (problem ) and unique -minimizer (problem ) for representations . This functional sparsity theorem thereby generalizes the DEGN result to a broader class of spaces, enabling sparse recovery in Banach-space settings and broadening potential applications in functional analysis and signal processing.

Abstract

Celebrated breakthrough sparsity theorem obtained independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]} and Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard -minimization problem can be obtained using unique solution to P-Type -minimization problem. In this paper, we extend their result to abstract Banach spaces using 1-approximate Schauder frames. We notice that the `normalized' condition for Hilbert spaces can be generalized to a larger extent when we consider Banach spaces.

Paper Structure

This paper contains 2 sections, 5 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Functional Donoho-Elad-Gribonval-Nielsen-Fuchs Sparsity Theorem

Key Result

Theorem 1.3

DONOHOELADGRIBONVALNIELSENELADKUTYNIOKFUCHSFUCHS2 (Donoho-Elad-Gribonval-Nielsen-Fuchs Sparsity Theorem) Let $\{\tau_j\}_{j=1}^n$ be a normalized frame for a Hilbert space $\mathcal{H}$. If $h \in \mathcal{H}$ can be written as $h=\theta_\tau^*c$ for some $c\in \mathbb{K}^n$ satisfying then $c$ is the unique solution to Problem P1 and Problem P0.

Theorems & Definitions (11)

  • Theorem 1.3
  • Definition 2.1
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • proof
  • Corollary 2.8
  • ...and 1 more