On denoising modulo 1 samples of a function

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 ) mod 1 for x_i  ∈ [0,1] , where η_i denotes noise. Given the samples (x_i ,y_i ) n i=1 , our goal is to recover smooth, robust estimates of the clean samples f(x_i) mod 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.