Signal recovery using Gabor frames

TL;DR

This paper introduces a probabilistic Gabor frame-based method for discrete signal recovery with random missing data, providing high-probability guarantees even beyond classical sparsity limits.

Contribution

It presents the first rigorous probabilistic recovery guarantee leveraging row-wise Gabor transform structure for signals with stochastic data loss.

Findings

Recovery probability approaches 1 as signal size increases

High-probability exact recovery under random missing frequencies

Novel maximal row-support criterion for unique reconstruction

Abstract

We present a novel probabilistic framework for the recovery of discrete signals with missing data, extending classical Fourier-based methods. While prior results, such as those of Donoho and Stark; see also Logan's method, guarantee exact recovery under strict deterministic sparsity constraints, they do not account for stochastic patterns of data loss. Our approach combines a row-wise Gabor transform with a probabilistic model for missing frequencies, establishing near-certain recovery when losses occur randomly. The key innovation is a maximal row-support criterion that allows unique reconstruction with high probability, even when the overall signal support significantly exceeds classical bounds. Specifically, we show that if missing frequencies are independently distributed according to a binomial law, the probability of exact recovery converges to $1$ as the signal size grows. This provides, to our knowledge, the first rigorous probabilistic recovery guarantee exploiting row-wise signal structure. Our framework offers new insights into the interplay between sparsity, transform structure, and stochastic loss, with immediate implications for communications, imaging, and data compression. It also opens avenues for future research, including extensions to higher-dimensional signals, adaptive transforms, and more general probabilistic loss models, potentially enabling even more robust recovery guarantees.