We revisit the classical problem of Fourier-sparse signal reconstruction -- a
variant of the \emph{Set Query} problem -- which asks to efficiently
reconstruct (a subset of) a d-dimensional Fourier-sparse signal
(β₯x^(t)β₯0ββ€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 (SFT) for frequencies lying on a
\emph{lattice}, 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:
β **Dimension-free Fourier sparse recovery** We present a
sample-optimal discrete Fourier Set-Query algorithm with O(kΟ+1)
reconstruction time in one dimension, \emph{independent} of the signal's length
(n) and βββ-norm. This complements the state-of-art algorithm of
[Kapralov, STOC 2017], whose reconstruction time is O~(klog2nlogRβ), where Rβββ₯x^β₯ββ is a signal-dependent parameter,
and the algorithm 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. A key component in our algorithm is fast spectral
sparsification of the Fourier basis.
β **High-accuracy Fourier interpolation** In one dimension, we design
a poly-time (3+2β+Ο΅)-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 main contribution is
a new analytic tool for hierarchical frequency decomposition based on
\emph{noise cancellation}