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Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity

Publication ,  Conference
Behne, JK; Reeves, G
Published in: Proceedings of Machine Learning Research
January 1, 2022

Low-rank matrix recovery problems involving high-dimensional and heterogeneous data appear in applications throughout statistics and machine learning. The contribution of this paper is to establish the fundamental limits of recovery for a broad class of these problems. In particular, we study the problem of estimating a rank-one matrix from Gaussian observations where different blocks of the matrix are observed under different noise levels. In the setting where the number of blocks is fixed while the number of variables tends to infinity, we prove asymptotically exact formulas for the minimum mean-squared error in estimating both the matrix and underlying factors. These results are based on a novel reduction from the low-rank matrix tensor product model (with homogeneous noise) to a rank-one model with heteroskedastic noise. As an application of our main result, we show that recently proposed methods based on applying principal component analysis (PCA) to weighted combinations of the data are optimal in some settings but sub-optimal in others. We also provide numerical results comparing our asymptotic formulas with the performance of methods based on weighted PCA, gradient descent, and approximate message passing.

Duke Scholars

Published In

Proceedings of Machine Learning Research

EISSN

2640-3498

Publication Date

January 1, 2022

Volume

151

Start / End Page

8650 / 8672
 

Citation

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Behne, J. K., & Reeves, G. (2022). Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity. In Proceedings of Machine Learning Research (Vol. 151, pp. 8650–8672).
Behne, J. K., and G. Reeves. “Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity.” In Proceedings of Machine Learning Research, 151:8650–72, 2022.
Behne JK, Reeves G. Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity. In: Proceedings of Machine Learning Research. 2022. p. 8650–72.
Behne, J. K., and G. Reeves. “Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity.” Proceedings of Machine Learning Research, vol. 151, 2022, pp. 8650–72.
Behne JK, Reeves G. Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity. Proceedings of Machine Learning Research. 2022. p. 8650–8672.

Published In

Proceedings of Machine Learning Research

EISSN

2640-3498

Publication Date

January 1, 2022

Volume

151

Start / End Page

8650 / 8672