Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-negative Sparse Recovery at Minimal Sampling Rate

Hendrik Bernd Zarucha, Peter Jung

TL;DR

The paper addresses uniform, robust recovery of non-negative $S$-sparse signals from minimal measurements by introducing the signed kernel condition, a non-null-space criterion enabling polynomial-time recovery via the non-negative least residual. It establishes formal equivalences between robustness and recovery, and extends these results to real Vandermonde constructions and complex formulations, showing that $M=O(S)$ measurements suffice under the signed kernel condition. It connects the recovery guarantees to outwardly neighborly polytopes and provides practical constructions, including linear-programming-based decoding for real cases. A bound on the complexity constant reveals the trade-off between dimension, sparsity, and measurements, highlighting regimes where near-optimal sampling is feasible and regimes where robustness constants deteriorate with problem size. Overall, the work fills a longstanding gap by achieving uniform, robust non-negative recovery with minimal sampling and polynomial-time decoding in structured settings.

Abstract

It is known that sparse recovery is possible if the number of measurements is in the order of the sparsity, but the corresponding decoders either lack polynomial decoding time or robustness to noise. Commonly, decoders that rely on a null space property are being used. These achieve polynomial time decoding and are robust to additive noise but pay the price by requiring more measurements. The non-negative least residual has been established as such a decoder for non-negative recovery. A new equivalent condition for uniform, robust recovery of non-negative sparse vectors with the non-negative least residual that is not based on null space properties is introduced. It is shown that the number of measurements for this equivalent condition only needs to be in the order of the sparsity. Further, it is explained why the robustness to additive noise is similar, but not equal, to the robustness of decoders based on null space properties.

Non-negative Sparse Recovery at Minimal Sampling Rate

TL;DR

The paper addresses uniform, robust recovery of non-negative -sparse signals from minimal measurements by introducing the signed kernel condition, a non-null-space criterion enabling polynomial-time recovery via the non-negative least residual. It establishes formal equivalences between robustness and recovery, and extends these results to real Vandermonde constructions and complex formulations, showing that measurements suffice under the signed kernel condition. It connects the recovery guarantees to outwardly neighborly polytopes and provides practical constructions, including linear-programming-based decoding for real cases. A bound on the complexity constant reveals the trade-off between dimension, sparsity, and measurements, highlighting regimes where near-optimal sampling is feasible and regimes where robustness constants deteriorate with problem size. Overall, the work fills a longstanding gap by achieving uniform, robust non-negative recovery with minimal sampling and polynomial-time decoding in structured settings.

Abstract

It is known that sparse recovery is possible if the number of measurements is in the order of the sparsity, but the corresponding decoders either lack polynomial decoding time or robustness to noise. Commonly, decoders that rely on a null space property are being used. These achieve polynomial time decoding and are robust to additive noise but pay the price by requiring more measurements. The non-negative least residual has been established as such a decoder for non-negative recovery. A new equivalent condition for uniform, robust recovery of non-negative sparse vectors with the non-negative least residual that is not based on null space properties is introduced. It is shown that the number of measurements for this equivalent condition only needs to be in the order of the sparsity. Further, it is explained why the robustness to additive noise is similar, but not equal, to the robustness of decoders based on null space properties.
Paper Structure (7 sections, 19 theorems, 83 equations, 1 table)

This paper contains 7 sections, 19 theorems, 83 equations, 1 table.

Table of Contents

  1. Introduction
  2. Robustness and Recovery Conditions
  3. The Signed Kernel Condition Case
  4. Real Signed Kernel Condition Case
  5. Complex Signed Kernel Condition Case
  6. Outwardly Neighborly Polytopes
  7. Bound for the Complexity Constants

Key Result

Theorem 2.2

Let $C\subset F\subset\mathbb{K}^N$, $\left\|\cdot\right\|_L$ a norm on $\mathbb{K}^N$, $\left\|\cdot\right\|_R$ a norm on $\mathbb{K}^M$ and $\mathbf{A}\in\mathbb{K}^{M\times N}$. Consider the following conditions: Then, the following statements are true:

Theorems & Definitions (49)

  • Definition 2.1
  • Theorem 2.2: Robustness and Recovery
  • proof
  • Theorem 2.3
  • proof
  • Example 2.4
  • Theorem 2.5: Measurement Matrix with bounded $\kappa$
  • proof
  • Theorem 2.6: Stability as a consequence of Robustness
  • proof
  • ...and 39 more