On Denoising Modulo 1 Samples of a Function

Abstract

Consider an unknown smooth function $f: [0,1] \rightarrow \mathbb{R}$, and say we are given $n$ noisy$\mod 1$ samples of $f$, i.e., $y_i = (f(x_i) + \eta_i)\mod 1$ for $x_i \in [0,1]$, where $\eta_i$ denotes noise. Given the samples $(x_i,y_i)_{i=1}^{n}$ our goal is to recover smooth, robust estimates of the clean samples $f(x_i) \bmod 1$. 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.