Deterministic Sparse Fourier Transform with an ?_{?} Guarantee

In this paper we revisit the deterministic version of the Sparse Fourier Transform problem, which asks to read only a few entries of $x \in \mathbb{C}^n$ and design a recovery algorithm such that the output of the algorithm approximates $\hat x$, the Discrete Fourier Transform (DFT) of $x$. The randomized case has been well-understood, while the main work in the deterministic case is that of Merhi et al.\@ (J Fourier Anal Appl 2018), which obtains $O(k^2 \log^{-1}k \cdot \log^{5.5}n)$ samples and a similar runtime with the $\ell_2/\ell_1$ guarantee. We focus on the stronger $\ell_{\infty}/\ell_1$ guarantee and the closely related problem of incoherent matrices. We list our contributions as follows. 1. We find a deterministic collection of $O(k^2 \log n)$ samples for the $\ell_\infty/\ell_1$ recovery in time $O(nk \log^2 n)$, and a deterministic collection of $O(k^2 \log^2 n)$ samples for the $\ell_\infty/\ell_1$ sparse recovery in time $O(k^2 \log^3n)$. 2. We give new deterministic constructions of incoherent matrices that are row-sampled submatrices of the DFT matrix, via a derandomization of Bernstein's inequality and bounds on exponential sums considered in analytic number theory. Our first construction matches a previous randomized construction of Nelson, Nguyen and Woodruff (RANDOM'12), where there was no constraint on the form of the incoherent matrix. Our algorithms are nearly sample-optimal, since a lower bound of $\Omega(k^2 + k \log n)$ is known, even for the case where the sensing matrix can be arbitrarily designed. A similar lower bound of $\Omega(k^2 \log n/ \log k)$ is known for incoherent matrices.Comment: ICALP 2020--presentation improved according to reviewers' comment

Sparse Recovery for Orthogonal Polynomial Transforms

In this paper we consider the following sparse recovery problem. We have query access to a vector ? ? ?^N such that x? = ? ? is k-sparse (or nearly k-sparse) for some orthogonal transform ?. The goal is to output an approximation (in an ?? sense) to x? in sublinear time. This problem has been well-studied in the special case that ? is the Discrete Fourier Transform (DFT), and a long line of work has resulted in sparse Fast Fourier Transforms that run in time O(k ? polylog N). However, for transforms ? other than the DFT (or closely related transforms like the Discrete Cosine Transform), the question is much less settled. In this paper we give sublinear-time algorithms - running in time poly(k log(N)) - for solving the sparse recovery problem for orthogonal transforms ? that arise from orthogonal polynomials. More precisely, our algorithm works for any ? that is an orthogonal polynomial transform derived from Jacobi polynomials. The Jacobi polynomials are a large class of classical orthogonal polynomials (and include Chebyshev and Legendre polynomials as special cases), and show up extensively in applications like numerical analysis and signal processing. One caveat of our work is that we require an assumption on the sparsity structure of the sparse vector, although we note that vectors with random support have this property with high probability. Our approach is to give a very general reduction from the k-sparse sparse recovery problem to the 1-sparse sparse recovery problem that holds for any flat orthogonal polynomial transform; then we solve this one-sparse recovery problem for transforms derived from Jacobi polynomials. Frequently, sparse FFT algorithms are described as implementing such a reduction; however, the technical details of such works are quite specific to the Fourier transform and moreover the actual implementations of these algorithms do not use the 1-sparse algorithm as a black box. In this work we give a reduction that works for a broad class of orthogonal polynomial families, and which uses any 1-sparse recovery algorithm as a black box