Deterministic Sparse FFT via Keyed Multi-View Gating with $O(\sqrt{N} \log k)$ Expected Time

TL;DR

This paper presents a deterministic sparse FFT algorithm using multi-view gating and CRT agreement, achieving expected sublinear time with probabilistic guarantees and worst-case safety.

Contribution

It introduces a deterministic candidate reduction method with a peeling recovery process, improving sparse FFT efficiency while maintaining worst-case guarantees.

Findings

Expected identification time is $O(\sqrt{N} \log k)$

Achieves deterministic candidate reduction without randomized bucketization

Provides probabilistic guarantees with no false negatives under verification

Abstract

We introduce a deterministic sparse Fourier transform framework based on a keyed multi-view gating mechanism that leverages 2-of-3 Chinese Remainder Theorem (CRT) agreement to reduce candidate frequency pairs from $O(k^2)$ to $\Theta(k)$ under sparse-regime assumptions. Unlike prior approaches that rely on randomized bucketization for candidate formation, the proposed method provides deterministic structure with probabilistic guarantees arising only from assumptions on frequency placement and independence of affine hashing across views. The algorithm is realized through a peeling-based recovery procedure that extracts frequencies directly from singleton bins without explicit pair enumeration. A recursive self-reduction eliminates the $O(\sqrt{N} \log N)$ preprocessing floor, yielding $O(\sqrt{N} \log k)$ expected identification time while maintaining an $O(N \log N)$ worst-case bound via deterministic dense-FFT fallback. A multi-view verification framework combining Parseval energy consistency and bin-wise residual checks ensures bounded failure probability and no false negatives under correct verification. This establishes a framework combining deterministic candidate reduction, sublinear expected complexity, and worst-case safety guarantees within a CRT-based sparse FFT architecture.