On denoising modulo 1 samples of a function

Consider an unknown smooth function f: [0,1] rightarrow ℝ, and say we are given n noisymod 1 samples of f, i.e., yi = (f(xi) + ηi)mod 1 for xi ∈ [0,1], where ηi denotes noise. Given the samples (xi,yi)i=1n our goal is to recover smooth, robust estimates of the clean samples f(xi) bmod 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.