Quartic Samples Suffice for Fourier Interpolation
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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 O(k^51) to O(k^4). ∙ The time complexity is improved from O(k^10ω+40) to O(k^4 ω). ∙ The output sparsity is improved from O(k^10) to O(k^4). Here, ω denotes the exponent of fast matrix multiplication. The state-of-the-art sample complexity of this problem is O(k), but can only be achieved by an *exponential-time* algorithm. Our algorithm uses slightly more samples (∼ k^4) in exchange for small polynomial runtime, laying the groundwork for a practical 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.
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