On denoising modulo 1 samples of a function
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Consider an unknown smooth function f: [0,1] →R, and say we are given n noisy mod 1 samples of f, i.e., y_i = (f(x_i) + η_i) 1 for x_i ∈ [0,1], where η_i denotes noise. Given the samples (x_i,y_i)_i=1^n, our goal is to recover smooth, robust estimates of the clean samples f(x_i) 1. We formulate a natural approach for solving this problem which works with angular embeddings of the noisy mod 1 samples over the unit complex circle, inspired by the angular synchronization framework. Our approach amounts to solving a quadratically constrained quadratic program (QCQP) which is NP-hard in its basic form, and therefore we consider its relaxation which is a trust region sub-problem and hence solvable efficiently. We demonstrate its robustness to noise via extensive numerical simulations on several synthetic examples, along with a detailed theoretical analysis. To the best of our knowledge, we provide the first algorithm for denoising mod 1 samples of a smooth function, which comes with robustness guarantees.
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