Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality

Changxiao Cai, H. Vincent Poor, Yuxin Chen
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:1271-1282, 2020.

Abstract

We study the distribution and uncertainty of nonconvex optimization for noisy tensor completion — the problem of estimating a low-rank tensor given incomplete and corrupted observations of its entries. Focusing on a two-stage nonconvex estimation algorithm proposed by (Cai et al., 2019), we characterize the distribution of this estimator down to fine scales. This distributional theory in turn allows one to construct valid and short confidence intervals for both the unseen tensor entries and its underlying tensor factors. The proposed inferential procedure enjoys several important features: (1) it is fully adaptive to noise heteroscedasticity, and (2) it is data-driven and adapts automatically to unknown noise distributions. Furthermore, our findings unveil the statistical optimality of nonconvex tensor completion: it attains un-improvable estimation accuracy — including both the rates and the pre-constants — under i.i.d. Gaussian noise.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-cai20c, title = {Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality}, author = {Cai, Changxiao and Poor, H. Vincent and Chen, Yuxin}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {1271--1282}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/cai20c/cai20c.pdf}, url = {https://proceedings.mlr.press/v119/cai20c.html}, abstract = {We study the distribution and uncertainty of nonconvex optimization for noisy tensor completion — the problem of estimating a low-rank tensor given incomplete and corrupted observations of its entries. Focusing on a two-stage nonconvex estimation algorithm proposed by (Cai et al., 2019), we characterize the distribution of this estimator down to fine scales. This distributional theory in turn allows one to construct valid and short confidence intervals for both the unseen tensor entries and its underlying tensor factors. The proposed inferential procedure enjoys several important features: (1) it is fully adaptive to noise heteroscedasticity, and (2) it is data-driven and adapts automatically to unknown noise distributions. Furthermore, our findings unveil the statistical optimality of nonconvex tensor completion: it attains un-improvable estimation accuracy — including both the rates and the pre-constants — under i.i.d. Gaussian noise.} }
Endnote
%0 Conference Paper %T Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality %A Changxiao Cai %A H. Vincent Poor %A Yuxin Chen %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-cai20c %I PMLR %P 1271--1282 %U https://proceedings.mlr.press/v119/cai20c.html %V 119 %X We study the distribution and uncertainty of nonconvex optimization for noisy tensor completion — the problem of estimating a low-rank tensor given incomplete and corrupted observations of its entries. Focusing on a two-stage nonconvex estimation algorithm proposed by (Cai et al., 2019), we characterize the distribution of this estimator down to fine scales. This distributional theory in turn allows one to construct valid and short confidence intervals for both the unseen tensor entries and its underlying tensor factors. The proposed inferential procedure enjoys several important features: (1) it is fully adaptive to noise heteroscedasticity, and (2) it is data-driven and adapts automatically to unknown noise distributions. Furthermore, our findings unveil the statistical optimality of nonconvex tensor completion: it attains un-improvable estimation accuracy — including both the rates and the pre-constants — under i.i.d. Gaussian noise.
APA
Cai, C., Poor, H.V. & Chen, Y.. (2020). Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:1271-1282 Available from https://proceedings.mlr.press/v119/cai20c.html.

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