We study the problem of interpolating a noisy Fourier-sparse signal in the time duration $[0, T]$ from noisy samples in the same range, where the ground truth signal can be any $k$-Fourier-sparse signal with band-limit $[-F, F]$. Our main result is an efficient Fourier Interpolation algorithm that improves the previous best algorithm by [Chen, Kane, Price, and Song, FOCS 2016] in the following three aspects: $\bullet$ The sample complexity is improved from $\widetilde{O}(k^{51})$ to $\widetilde{O}(k^{4})$. $\bullet$ The time complexity is improved from $ \widetilde{O}(k^{10\omega+40})$ to $\widetilde{O}(k^{4 \omega})$. $\bullet$ The output sparsity is improved from $\widetilde{O}(k^{10})$ to $\widetilde{O}(k^{4})$. Here, $\omega$ denotes the exponent of fast matrix multiplication. The state-of-the-art sample complexity of this problem is $\sim k^4$, but was only known to be achieved by an *exponential-time* algorithm. Our algorithm uses the same number of samples but has a polynomial runtime, laying the groundwork for an efficient Fourier Interpolation algorithm. The centerpiece of our algorithm is a new sufficient condition for the frequency estimation task -- a high signal-to-noise (SNR) band condition -- which allows for efficient and accurate signal reconstruction. Based on this condition together with a new structural decomposition of Fourier signals (Signal Equivalent Method), we design a cheap algorithm for estimating each "significant" frequency within a narrow range, which is then combined with a signal estimation algorithm into a new Fourier Interpolation framework to reconstruct the ground-truth signal.

翻译：我们研究在时间区间 $[0, T]$ 内，从带噪样本中插值带噪傅里叶稀疏信号的问题，其中真实信号可以是任意带限为 $[-F, F]$ 的 $k$-傅里叶稀疏信号。我们的主要成果是一种高效的傅里叶插值算法，它在以下三个方面改进了 [Chen, Kane, Price, and Song, FOCS 2016] 之前的最优算法： $\bullet$ 样本复杂度从 $\widetilde{O}(k^{51})$ 提升至 $\widetilde{O}(k^{4})$。 $\bullet$ 时间复杂度从 $\widetilde{O}(k^{10\omega+40})$ 提升至 $\widetilde{O}(k^{4 \omega})$。 $\bullet$ 输出稀疏度从 $\widetilde{O}(k^{10})$ 提升至 $\widetilde{O}(k^{4})$。 这里 $\omega$ 表示快速矩阵乘法的指数。该问题当前最优的样本复杂度为 $\sim k^4$，但此前仅能通过*指数时间*算法实现。我们的算法使用相同数量的样本，却具有多项式运行时间，为高效傅里叶插值算法奠定了坚实基础。 本算法的核心是频率估计任务的一个新充分条件——高信噪比（SNR）频带条件——它能够实现高效且精确的信号重构。基于该条件以及傅里叶信号的新结构分解（信号等价方法），我们设计了一种廉价算法，用于在窄范围内估计每个“显著”频率，随后将其与信号估计算法相结合，形成一种新的傅里叶插值框架，以重构真实信号。