On denoising modulo 1 samples of a function
This addresses a specific denoising problem in signal processing or data analysis, but appears incremental as it builds on existing QCQP and relaxation methods.
The paper tackles the problem of recovering smooth estimates from noisy modulo 1 samples of a function by formulating it as a QCQP with non-convex constraints and solving it via an efficient trust-region relaxation. The approach demonstrates robustness to noise in simulations and includes theoretical analysis.
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., $y_i = (f(x_i) + η_i)\mod 1$ for $x_i \in [0,1]$, where $η_i$ 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.