On denoising modulo 1 samples of a function

Mihai Cucuringu, Hemant Tyagi
Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, PMLR 84:1868-1876, 2018.

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.

Cite this Paper

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. } }
Endnote
%0 Conference Paper %T On denoising modulo 1 samples of a function %A Mihai Cucuringu %A Hemant Tyagi %B Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2018 %E Amos Storkey %E Fernando Perez-Cruz %F pmlr-v84-cucuringu18a %I PMLR %P 1868--1876 %U https://proceedings.mlr.press/v84/cucuringu18a.html %V 84 %X 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.
APA
Cucuringu, M. & Tyagi, H.. (2018). On denoising modulo 1 samples of a function. Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 84:1868-1876 Available from https://proceedings.mlr.press/v84/cucuringu18a.html.

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