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Register a new userTowards Faster Rates and Oracle Property for Low-Rank Matrix Estimation
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Added by
Quanquan Gu
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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We present a unified framework for low-rank matrix estimation with nonconvex penalties. We first prove that the proposed estimator attains a faster statistical rate than the traditional low-rank matrix estimator with nuclear norm penalty. Moreover, we rigorously show that under a certain condition on the magnitude of the nonzero singular values, the proposed estimator enjoys oracle property (i.e., exactly recovers the true rank of the matrix), besides attaining a faster rate. As far as we know, this is the first work that establishes the theory of low-rank matrix estimation with nonconvex penalties, confirming the advantages of nonconvex penalties for matrix completion. Numerical experiments on both synthetic and real world datasets corroborate our theory.
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Existing results for low-rank matrix recovery largely focus on quadratic loss, which enjoys favorable properties such as restricted strong convexity/smoothness (RSC/RSM) and well conditioning over all low rank matrices. However, many interesting problems involve non-quadratic loss do not satisfy such properties; examples including one-bit matrix sensing, one-bit matrix completion, and rank aggregation. For these problems, standard nonconvex approaches such as projected gradient with rank constraint alone (a.k.a. iterative hard thresholding) and Burer-Monteiro approach may perform badly in practice and have no satisfactory theory in guaranteeing global and efficient convergence. In this paper, we show that the critical component in low-rank recovery with non-quadratic loss is a regularity projection oracle, which restricts iterates to low-rank matrix within an appropriate bounded set, over which the loss function is well behaved and satisfies a set of relaxed RSC/RSM conditions. Accordingly, we analyze an (averaged) projected gradient method equipped with such an oracle, and prove that it converges globally and linearly. Our results apply to a wide range of non-quadratic problems including rank aggregation, one bit matrix sensing/completion, and more broadly generalized linear models with rank constraint.
Various problems in data analysis and statistical genetics call for recovery of a column-sparse, low-rank matrix from noisy observations. We propose ReFACTor, a simple variation of the classical Truncated Singular Value Decomposition (TSVD) algorithm. In contrast to previous sparse principal component analysis (PCA) algorithms, our algorithm can provably reveal a low-rank signal matrix better, and often significantly better, than the widely used TSVD, making it the algorithm of choice whenever column-sparsity is suspected. Empirically, we observe that ReFACTor consistently outperforms TSVD even when the underlying signal is not sparse, suggesting that it is generally safe to use ReFACTor instead of TSVD and PCA. The algorithm is extremely simple to implement and its running time is dominated by the runtime of PCA, making it as practical as standard principal component analysis.
Matrix completion is a modern missing data problem where both the missing structure and the underlying parameter are high dimensional. Although missing structure is a key component to any missing data problems, existing matrix completion methods often assume a simple uniform missing mechanism. In this work, we study matrix completion from corrupted data under a novel low-rank missing mechanism. The probability matrix of observation is estimated via a high dimensional low-rank matrix estimation procedure, and further used to complete the target matrix via inverse probabilities weighting. Due to both high dimensional and extreme (i.e., very small) nature of the true probability matrix, the effect of inverse probability weighting requires careful study. We derive optimal asymptotic convergence rates of the proposed estimators for both the observation probabilities and the target matrix.
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters. Such bounds are established for several statistically relevant distances, including quant
In this paper, we propose three approaches for the estimation of the Tucker decomposition of multi-way arrays (tensors) from partial observations. All approaches are formulated as convex minimization problems. Therefore, the minimum is guaranteed to be unique. The proposed approaches can automatically estimate the number of factors (rank) through the optimization. Thus, there is no need to specify the rank beforehand. The key technique we employ is the trace norm regularization, which is a popular approach for the estimation of low-rank matrices. In addition, we propose a simple heuristic to improve the interpretability of the obtained factorization. The advantages and disadvantages of three proposed approaches are demonstrated through numerical experiments on both synthetic and real world datasets. We show that the proposed convex optimization based approaches are more accurate in predictive performance, faster, and more reliable in recovering a known multilinear structure than conventional approaches.
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