Signal recovery using Gabor frames
Ivan Bortnovskyi, June Duvivier, Xiaoyao Huang, Alex Iosevich, Say-Yeon Kwon, Meiling Laurence, Michael Lucas, Steven J. Miller, Tiancheng Pan, Eyvindur Palsson, Jennifer Smucker, Iana Vranesko
TL;DR
The paper addresses recovery of discrete signals $f:\mathbb{Z}_N\times\mathbb{Z}_T\to\mathbb{C}$ under randomly missing data, extending classical deterministic sparsity bounds from the Donoho–Stark framework. It introduces a row-wise Gabor transform framework that exploits per-row sparsity to achieve probabilistic recovery guarantees rather than worst-case guarantees. The main results show that if missing frequencies follow an independent Bernoulli model with parameter $\theta$ satisfying $\theta<1/(2E_{\max})$, then the probability of exact recovery tends to $1$ as $N\to\infty$; when $\theta>1/(2E_{\max})$, analogous probabilities decay to zero. These results illuminate how transform structure and row-wise sparse supports interact with stochastic loss, yielding practical implications for communications, imaging, and data compression, and point to future directions including higher-dimensional extensions and adaptive transforms.
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.