Quartic Samples Suffice for Fourier Interpolation

• Published in: Proceedings / annual Symposium on Foundations of Computer Science pp. 1414 - 1425
• Main Authors: Song, Zhao, Sun, Baocheng, Weinstein, Omri, Zhang, Ruizhe
• Format: Conference Proceeding
• Language: English
• Published: IEEE 06.11.2023
• Subjects: Complexity theory; Estimation; Frequency estimation; Interpolation; Noise measurement; Runtime; Sufficient conditions
• ISSN: 2575-8454
• DOI: 10.1109/FOCS57990.2023.00087

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.