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 ($\|\hat{x}(t)\|_0 \leq k$), 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 (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: $\bullet$ **Dimension-free 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. This complements the state-of-art algorithm of [Kapralov, STOC 2017], whose reconstruction time is $\tilde{O}(k \log^2 n \log R^*)$, where $R^* \approx \|\hat{x}\|_\infty$ 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. $\bullet$ **High-accuracy Fourier interpolation** In one dimension, we design a poly-time $(3+ \sqrt{2} +\epsilon)$-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 main contribution is a new analytic tool for hierarchical frequency decomposition based on \emph{noise cancellation}.

翻译：我们重新审视了傅里叶稀疏信号重建这一经典问题——这是集合查询问题的一个变体——目标是从时域中 $x(t)$ 的最少噪声样本中高效重建一个 $d$ 维傅里叶稀疏信号（$\|\hat{x}(t)\|_0 \leq k$）的（子集）。我们通过发展频率位于晶格上的稀疏傅里叶变换理论，为该问题提出了一个统一框架，该理论可视为离散域与连续域之间稀疏傅里叶变换的“半连续”版本。利用这一框架，我们获得了以下结果： • **无维数依赖的傅里叶稀疏恢复**：我们提出了一种一维中样本最优的离散傅里叶集合查询算法，其重建时间为 $O(k^{\omega+1})$，与信号长度（$n$）和 $\ell_\infty$ 范数无关。这补充了 [Kapralov, STOC 2017] 的现有最优算法，后者的重建时间为 $\tilde{O}(k \log^2 n \log R^*)$，其中 $R^* \approx \|\hat{x}\|_\infty$ 是信号相关参数，且该算法仅限于低维。相比之下，我们的算法适用于任意 $d$ 维，将解码时间的 $\exp(d)$ 爆炸降低为与 $d$ 成线性关系。算法的关键组件是傅里叶基的快速谱稀疏化。 • **高精度傅里叶插值**：在一维中，我们设计了一个多项式时间的 $(3+ \sqrt{2} +\epsilon)$ 近似算法用于连续傅里叶插值。这突破了之前所有算法 [Price and Song, FOCS 2015, Chen, Kane, Price and Song, FOCS 2016] 的障碍，这些算法对此基本问题仅能达到 $c>100$ 的近似比。我们的主要贡献是基于噪声消除的层次化频率分解新分析工具。