Sparse Fourier Transform over Lattices: A Unified Approach to Signal Reconstruction

We revisit the classical problem of band-limited signal reconstruction -- a variant of the \emph{Set Query} problem -- which asks to efficiently reconstruct (a subset of) a $d$-dimensional Fourier-sparse signal ($\|\widehat{x}(t)\|_0 \leq k$), from minimum \emph{noisy} samples of $x(t)$ in the time domain. We present a unified framework for this problem, by developing a theory of sparse Fourier transforms over \emph{lattices}, which can be viewed as a ``semi-continuous'' version of SFT, in-between discrete and continuous domains. Using this framework, we obtain the following results: $\bullet$ *High-dimensional Fourier sparse recovery* We present a sample-optimal discrete Fourier Set-Query algorithm with $O(k^{\omega+1})$ reconstruction time in one dimension, \emph{independent} of the signal's length ($n$) and $\ell_\infty$-norm ($R^* \approx \|\widehat{x}\|_\infty$). This complements the state-of-art algorithm of [Kapralov, STOC 2017], whose reconstruction time is $\tilde{O}(k \log^2 n \log R^*)$, and is limited to low-dimensions. By contrast, our algorithm works for arbitrary $d$ dimensions, mitigating the $\exp(d)$ blowup in decoding time to merely linear in $d$. Our algorithm also works for the semi-continuous case where frequencies lie on a lattice. $\bullet$ *High-accuracy Fourier interpolation* We design a polynomial-time $(1+ \sqrt{2} +\epsilon)$-approximation algorithm for continuous Fourier interpolation. This bypasses a barrier of all previous algorithms [Price and Song, FOCS 2015, Chen, Kane, Price and Song, FOCS 2016] which only achieve $c>100$ approximation for this basic problem. Our algorithm relies on several new ideas of independent interests in signal estimation, including high-sensitivity frequency estimation and new error analysis with sharper noise control.