Homomorphic Sensing: Sparsity and Noise

Liangzu Peng, Boshi Wang, Manolis Tsakiris
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:8464-8475, 2021.

Abstract

\emph{Unlabeled sensing} is a recent problem encompassing many data science and engineering applications and typically formulated as solving linear equations whose right-hand side vector has undergone an unknown permutation. It was generalized to the \emph{homomorphic sensing} problem by replacing the unknown permutation with an unknown linear map from a given finite set of linear maps. In this paper we present tighter and simpler conditions for the homomorphic sensing problem to admit a unique solution. We show that this solution is locally stable under noise, while under a sparsity assumption it remains unique under less demanding conditions. Sparsity in the context of unlabeled sensing leads to the problem of \textit{unlabeled compressed sensing}, and a consequence of our general theory is the existence under mild conditions of a unique sparsest solution. On the algorithmic level, we solve unlabeled compressed sensing by an iterative algorithm validated by synthetic data experiments. Finally, under the unifying homomorphic sensing framework we connect unlabeled sensing to other important practical problems.

Cite this Paper


BibTeX
@InProceedings{pmlr-v139-peng21a, title = {Homomorphic Sensing: Sparsity and Noise}, author = {Peng, Liangzu and Wang, Boshi and Tsakiris, Manolis}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {8464--8475}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/peng21a/peng21a.pdf}, url = {http://proceedings.mlr.press/v139/peng21a.html}, abstract = {\emph{Unlabeled sensing} is a recent problem encompassing many data science and engineering applications and typically formulated as solving linear equations whose right-hand side vector has undergone an unknown permutation. It was generalized to the \emph{homomorphic sensing} problem by replacing the unknown permutation with an unknown linear map from a given finite set of linear maps. In this paper we present tighter and simpler conditions for the homomorphic sensing problem to admit a unique solution. We show that this solution is locally stable under noise, while under a sparsity assumption it remains unique under less demanding conditions. Sparsity in the context of unlabeled sensing leads to the problem of \textit{unlabeled compressed sensing}, and a consequence of our general theory is the existence under mild conditions of a unique sparsest solution. On the algorithmic level, we solve unlabeled compressed sensing by an iterative algorithm validated by synthetic data experiments. Finally, under the unifying homomorphic sensing framework we connect unlabeled sensing to other important practical problems.} }
Endnote
%0 Conference Paper %T Homomorphic Sensing: Sparsity and Noise %A Liangzu Peng %A Boshi Wang %A Manolis Tsakiris %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-peng21a %I PMLR %P 8464--8475 %U http://proceedings.mlr.press/v139/peng21a.html %V 139 %X \emph{Unlabeled sensing} is a recent problem encompassing many data science and engineering applications and typically formulated as solving linear equations whose right-hand side vector has undergone an unknown permutation. It was generalized to the \emph{homomorphic sensing} problem by replacing the unknown permutation with an unknown linear map from a given finite set of linear maps. In this paper we present tighter and simpler conditions for the homomorphic sensing problem to admit a unique solution. We show that this solution is locally stable under noise, while under a sparsity assumption it remains unique under less demanding conditions. Sparsity in the context of unlabeled sensing leads to the problem of \textit{unlabeled compressed sensing}, and a consequence of our general theory is the existence under mild conditions of a unique sparsest solution. On the algorithmic level, we solve unlabeled compressed sensing by an iterative algorithm validated by synthetic data experiments. Finally, under the unifying homomorphic sensing framework we connect unlabeled sensing to other important practical problems.
APA
Peng, L., Wang, B. & Tsakiris, M.. (2021). Homomorphic Sensing: Sparsity and Noise. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:8464-8475 Available from http://proceedings.mlr.press/v139/peng21a.html.

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