Exponential Family Matrix Completion under Structural Constraints

Suriya Gunasekar, Pradeep Ravikumar, Joydeep Ghosh
Proceedings of the 31st International Conference on Machine Learning, PMLR 32(2):1917-1925, 2014.

Abstract

We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low–rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin–tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data–types, such as skewed–continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low–rank, such as block–sparsity, or a superposition structure of low–rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of \textitexponential family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer \mathcalR(.). We propose a simple convex regularized M–estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v32-gunasekar14, title = {Exponential Family Matrix Completion under Structural Constraints}, author = {Gunasekar, Suriya and Ravikumar, Pradeep and Ghosh, Joydeep}, booktitle = {Proceedings of the 31st International Conference on Machine Learning}, pages = {1917--1925}, year = {2014}, editor = {Xing, Eric P. and Jebara, Tony}, volume = {32}, number = {2}, series = {Proceedings of Machine Learning Research}, address = {Bejing, China}, month = {22--24 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v32/gunasekar14.pdf}, url = {https://proceedings.mlr.press/v32/gunasekar14.html}, abstract = {We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low–rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin–tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data–types, such as skewed–continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low–rank, such as block–sparsity, or a superposition structure of low–rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of \textitexponential family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer \mathcalR(.). We propose a simple convex regularized M–estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets.} }
Endnote
%0 Conference Paper %T Exponential Family Matrix Completion under Structural Constraints %A Suriya Gunasekar %A Pradeep Ravikumar %A Joydeep Ghosh %B Proceedings of the 31st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2014 %E Eric P. Xing %E Tony Jebara %F pmlr-v32-gunasekar14 %I PMLR %P 1917--1925 %U https://proceedings.mlr.press/v32/gunasekar14.html %V 32 %N 2 %X We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low–rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin–tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data–types, such as skewed–continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low–rank, such as block–sparsity, or a superposition structure of low–rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of \textitexponential family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer \mathcalR(.). We propose a simple convex regularized M–estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets.
RIS
TY - CPAPER TI - Exponential Family Matrix Completion under Structural Constraints AU - Suriya Gunasekar AU - Pradeep Ravikumar AU - Joydeep Ghosh BT - Proceedings of the 31st International Conference on Machine Learning DA - 2014/06/18 ED - Eric P. Xing ED - Tony Jebara ID - pmlr-v32-gunasekar14 PB - PMLR DP - Proceedings of Machine Learning Research VL - 32 IS - 2 SP - 1917 EP - 1925 L1 - http://proceedings.mlr.press/v32/gunasekar14.pdf UR - https://proceedings.mlr.press/v32/gunasekar14.html AB - We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low–rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin–tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data–types, such as skewed–continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low–rank, such as block–sparsity, or a superposition structure of low–rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of \textitexponential family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer \mathcalR(.). We propose a simple convex regularized M–estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets. ER -
APA
Gunasekar, S., Ravikumar, P. & Ghosh, J.. (2014). Exponential Family Matrix Completion under Structural Constraints. Proceedings of the 31st International Conference on Machine Learning, in Proceedings of Machine Learning Research 32(2):1917-1925 Available from https://proceedings.mlr.press/v32/gunasekar14.html.

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