Sparse Fourier Transform over Lattices: A Unified Approach to Signal Reconstruction
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We revisit the classical problem of band-limited signal reconstruction – a variant of the *Set Query* problem – which asks to efficiently reconstruct (a subset of) a d-dimensional Fourier-sparse signal (x̂(t)_0 ≤ k), from minimum 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 over *lattices*, 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: ∙ *High-dimensional Fourier sparse recovery* We present a sample-optimal discrete Fourier Set-Query algorithm with O(k^ω+1) reconstruction time in one dimension, independent of the signal's length (n) and ℓ_∞-norm (R^* ≈x̂_∞). This complements the state-of-art algorithm of [Kap17], whose reconstruction time is Õ(k log^2 n log R^*), and 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. ∙ *High-accuracy Fourier interpolation* We design a polynomial-time (1+ √(2) +ϵ)-approximation algorithm for continuous Fourier interpolation. This bypasses a barrier of all previous algorithms [PS15, CKPS16] which only achieve >100 approximation for this problem. Our algorithm relies on several new ideas of independent interest in signal estimation, including high-sensitivity frequency estimation and new error analysis with sharper noise control. ∙ *Fourier-sparse interpolation with optimal output sparsity* We give a k-Fourier-sparse interpolation algorithm with optimal output signal sparsity, improving on the approximation ratio, sample complexity and runtime of prior works [CKPS16, CP19].
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