On denoising modulo 1 samples of a function
Abstract
Consider an unknown smooth function $f: [0,1] →\mathbb{R}$, and say we are given $n$ noisy $\mod 1$ samples of $f$, i.e., $y_i = (f(x_i) + \eta_i)\mod 1$ for $x_i ∈[0,1]$, 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 subproblem, and hence solvable efficiently. We demonstrate its robustness to noise via extensive simulations on several synthetic examples, and provide a detailed theoretical analysis.
BibTeX
@InProceedings{pmlr-v84-cucuringu18a,
title = {On denoising modulo 1 samples of a function},
author = {Cucuringu, Mihai and Tyagi, Hemant},
booktitle = {Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics},
pages = {1868--1876},
year = {2018},
editor = {Storkey, Amos and Perez-Cruz, Fernando},
volume = {84},
series = {Proceedings of Machine Learning Research},
month = {09--11 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v84/cucuringu18a/cucuringu18a.pdf},
url = {https://proceedings.mlr.press/v84/cucuringu18a.html},
abstract = {Consider an unknown smooth function $f: [0,1] →\mathbb{R}$, and say we are given $n$ noisy $\mod 1$ samples of $f$, i.e., $y_i = (f(x_i) + \eta_i)\mod 1$ for $x_i ∈[0,1]$, 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 subproblem, and hence solvable efficiently. We demonstrate its robustness to noise via extensive simulations on several synthetic examples, and provide a detailed theoretical analysis. }
}