Uncertainty Principles on Finite Abelian Groups, Restriction Theory, and Applications to Sparse Signal Recovery

Abstract: Let $G$ be a finite abelian group. Let $f: G \to {\mathbb C}$ be a signal (i.e. function). The classical uncertainty principle asserts that the product of the size of the support of $f$ and its Fourier transform $\hat f$, $\text{supp}(f)$ and $\text{supp}(\hat f)$ respectively, must satisfy the condition: $$|\text{supp}(f)| \cdot |\text{supp}(\hat f)| \geq |G|.$$ In the first part of this paper, we improve the uncertainty principle for signals with Fourier transform supported on generic sets. This improvement is achieved by employing {\it the restriction theory} and {\it the Salem set} mechanism from harmonic analysis. Then we investigate some applications of uncertainty principles that were developed in the first part of this paper, to the problem of unique recovery of finite sparse signals in the absence of some frequencies. Donoho and Stark (\cite{DS89}) showed that a signal of length $N$ can be recovered exactly, even if some of the frequencies are unobserved, provided that the product of the size of the number of non-zero entries of the signal and the number of missing frequencies is not too large, leveraging the classical uncertainty principle for vectors. Our results broaden the scope for a natural class of signals in higher-dimensional spaces. In the case when the signal is binary, we provide a very simple exact recovery mechanism through the DRA algorithm.