On denoising modulo 1 samples of a function - Cucuringu, M; Tyagi, H (Proceedings of Machine Learning Research, 2018)
Consider an unknown smooth function f : [0, 1] → R, and say we are given n noisy mod 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) n i=1 our goal is to recover smooth, robust estimates of the clean samples f(xi) mod 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 subproblem, and hence solvable efficiently. We demonstrate its robustness to noise via extensive simulations on several synthetic examples, and provide a detailed theoretical analysis.