Quartic Samples Suffice for Fourier Interpolation

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:•The sample complexity is improved from $\widetilde{O}\left(k^{51}\right)$ to $\widetilde{O}\left(k^{4}\right)$.•The time complexity is improved from $\widetilde{O}\left(k^{10 \omega+40}\right)$ to $\widetilde{O}\left(k^{4 \omega}\right)$.•The output sparsity is improved from $\widetilde{O}\left(k^{10}\right)$ to $\widetilde{O}\left(k^{4}\right)$. 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 spectral analysis tool-the Signal Equivalent Method-which utilizes the structure of Fourier signals to establish nearly-optimal energy properties, and is the key for efficient and accurate frequency estimation. We use this method, along with a new sufficient condition for frequency recovery (a new high SNR band condition), to design a cheap algorithm for estimating “significant” frequencies within a narrow range. Together with a signal estimation algorithm, we obtain a new Fourier Interpolation algorithm for reconstructing the ground-truth signal.