Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Additive energy, uncertainty principle and signal recovery mechanisms

K. Aldahleh, A. Iosevich, J. Iosevich, J. Jaimangal, A. Mayeli, S. Pack

TL;DR

The paper develops an additive-energy framework to sharpen the classical Fourier-uncertainty-based recovery of signals from incomplete frequency data on finite abelian groups. By introducing the additive energy $\Lambda(A)$ and proving an additive-energy uncertainty principle, it yields improved recovery conditions over the Donoho–Stark bound, expressed in terms of $|E|$, $|S|$, and energy quantities of the supports. It then leverages these combinatorial-energy bounds to establish two actionable recovery results via $L^1$ and $L^2$ minimization: precise thresholds guaranteeing exact recovery from partial Fourier information when the missing-frequency set has controlled additive energy. The results connect additive combinatorics with practical signal reconstruction, offering sharper criteria and broadening the toolkit for exact recovery under partial data on $\mathbb{Z}_N^d$.

Abstract

Given a signal $f:G\to\mathbb{C}$, where $G$ is a finite abelian group, under what reasonable assumptions can we guarantee the exact recovery of $f$ from a proper subset of its Fourier coefficients? In 1989, Donoho and Stark established a result \cite{DS89} using the classical uncertainty principle, which states that $|\text{supp}(f)|\cdot|\text{supp}(\hat{f})|\geq |G|$ for any nonzero signal $f$. Another result, first proven by Santose and Symes \cite{SS86}, was based on the Logan phenomenon \cite{L65}. In particular, the result showcases how the $L^1$ and $L^2$ minimizing signals with matching Fourier frequencies often recovers the original signal. The purpose of this paper is to relate these recovery mechanisms to additive energy, a combinatorial measure denoted and defined by $$Λ(A)=\left| \left\{ (x_1, x_2, x_3, x_4) \in A^4 \mid x_1 + x_2 = x_3 + x_4 \right\} \right|,$$ where $A\subset\mathbb{Z}_N^d$. In the first part of this paper, we use combinatorial techniques to establish an improved variety of the uncertainty principle in terms of additive energy. In a similar fashion as the Donoho-Stark argument, we use this principle to establish an often stronger recovery condition. In the latter half of the paper, we invoke these combinatorial methods to demonstrate two $L^p$ minimizing recovery results.

Additive energy, uncertainty principle and signal recovery mechanisms

TL;DR

The paper develops an additive-energy framework to sharpen the classical Fourier-uncertainty-based recovery of signals from incomplete frequency data on finite abelian groups. By introducing the additive energy and proving an additive-energy uncertainty principle, it yields improved recovery conditions over the Donoho–Stark bound, expressed in terms of , , and energy quantities of the supports. It then leverages these combinatorial-energy bounds to establish two actionable recovery results via and minimization: precise thresholds guaranteeing exact recovery from partial Fourier information when the missing-frequency set has controlled additive energy. The results connect additive combinatorics with practical signal reconstruction, offering sharper criteria and broadening the toolkit for exact recovery under partial data on .

Abstract

Given a signal , where is a finite abelian group, under what reasonable assumptions can we guarantee the exact recovery of from a proper subset of its Fourier coefficients? In 1989, Donoho and Stark established a result \cite{DS89} using the classical uncertainty principle, which states that for any nonzero signal . Another result, first proven by Santose and Symes \cite{SS86}, was based on the Logan phenomenon \cite{L65}. In particular, the result showcases how the and minimizing signals with matching Fourier frequencies often recovers the original signal. The purpose of this paper is to relate these recovery mechanisms to additive energy, a combinatorial measure denoted and defined by where . In the first part of this paper, we use combinatorial techniques to establish an improved variety of the uncertainty principle in terms of additive energy. In a similar fashion as the Donoho-Stark argument, we use this principle to establish an often stronger recovery condition. In the latter half of the paper, we invoke these combinatorial methods to demonstrate two minimizing recovery results.

Paper Structure

This paper contains 7 sections, 12 theorems, 91 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Additive energy uncertainty principle
  3. Good energy and bad energy
  4. The $L^1$ and $L^2$ minimization signal recovery algorithms
  5. Proof of Theorem \ref{['energyuncertaintyprincipletheorem']}
  6. Proof of Theorem \ref{['theorem:L1minimization']}
  7. Proof of Theorem \ref{['theorem:L2minimization']}

Key Result

Theorem 1.1

Let $f: {\mathbb Z}_N^d \to \Bbb C$ be a finite signal in ${\mathbb Z}_N^d$ with $N_t\in \Bbb N$ non-zero entries. Suppose that the set of unobserved frequencies $\{\hat{f}(m)\}_{m\in \mathbb Z_N}$ is of size $N_w\in \Bbb N$. Then the signal $f$ can be recovered uniquely from the observed frequencie

Figures (4)

  • Figure 1: The discrete Fourier transform of a random set.
  • Figure 2: The discrete Fourier transform of an arithmetic progression.
  • Figure 3: The discrete Fourier transform of the union of an arithmetic progression and a random set.
  • Figure 4: The discrete Fourier transform of a union of an arithmetic progression and a random set, with the random part of the set much larger in size.

Theorems & Definitions (19)

  • Theorem 1.1: DS89
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Definition 1.6: Additive Energy
  • Proposition 1.7
  • Theorem 1.8: Additive Uncertainty Principle
  • Remark 1.9
  • Theorem 1.10: Theorem 3.12, IM24
  • ...and 9 more