Low-Rank Matrix Estimation From Rank-One Projections by Unlifted Convex Optimization

Sohail Bahmani; Kiryung Lee

Low-Rank Matrix Estimation From Rank-One Projections by Unlifted Convex Optimization

Statistics Theory 2021-01-12 v2 Machine Learning Machine Learning Statistics Theory

Authors: Sohail Bahmani , Kiryung Lee

Abstract

We study an estimator with a convex formulation for recovery of low-rank matrices from rank-one projections. Using initial estimates of the factors of the target $d_1\times d_2$ matrix of rank- $r$ , the estimator admits a practical subgradient method operating in a space of dimension $r(d_1+d_2)$ . This property makes the estimator significantly more scalable than the convex estimators based on lifting and semidefinite programming. Furthermore, we present a streamlined analysis for exact recovery under the real Gaussian measurement model, as well as the partially derandomized measurement model by using the spherical $t$ -design. We show that under both models the estimator succeeds, with high probability, if the number of measurements exceeds $r^2 (d_1+d_2)$ up to some logarithmic factors. This sample complexity improves on the existing results for nonconvex iterative algorithms.

Keywords

low-rank matrix approximation convex optimization covariance estimation

Cite

@article{arxiv.2004.02718,
  title  = {Low-Rank Matrix Estimation From Rank-One Projections by Unlifted Convex Optimization},
  author = {Sohail Bahmani and Kiryung Lee},
  journal= {arXiv preprint arXiv:2004.02718},
  year   = {2021}
}

Low-Rank Matrix Estimation From Rank-One Projections by Unlifted Convex Optimization

Abstract

Keywords

Cite

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