On denoising modulo 1 samples of a function

Abstract

Consider an unknown smooth function f: [0,1] R, and say we are given n noisy 1 samples of f, i.e., y_i = (f(x_i) + _i) 1 for x_i [0,1], where _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 representations of mod 1 values over the unit circle. This amounts to solving a quadratically constrained quadratic program (QCQP) with non-convex constraints involving points lying on the unit circle. Our proposed approach is based on solving its relaxation which is a trust-region sub-problem, and hence solvable efficiently. We demonstrate its robustness to noise % of our approach via extensive simulations on several synthetic examples, and provide a detailed theoretical analysis.

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