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Submitted to the Annals of Statistics arXiv: arXiv:1605.00353

RATE-OPTIMAL PERTURBATION BOUNDS FOR SINGULAR SUBSPACES WITH APPLICATIONS TO

HIGH-DIMENSIONAL STATISTICS

By T. Tony Cai∗ and Anru Zhang

University of Pennsylvania and University of Wisconsin-Madison

Perturbation bounds for singular spaces, in particular Wedin’s sin Θ theorem, are a fundamental tool in many fields including high- dimensional statistics, machine learning, and applied mathematics. In this paper, we establish separate perturbation bounds, measured in both spectral and Frobenius sin Θ distances, for the left and right singular subspaces. Lower bounds, which show that the individual perturbation bounds are rate-optimal, are also given.

The new perturbation bounds are applicable to a wide range of problems. In this paper, we consider in detail applications to low-rank matrix denoising and singular space estimation, high-dimensional clustering, and canonical correlation analysis (CCA). In particular, separate matching upper and lower bounds are obtained for estimat- ing the left and right singular spaces. To the best of our knowledge, this is the first result that gives different optimal rates for the left and right singular spaces under the same perturbation.

1. Introduction. Singular value decomposition (SVD) and spectral meth- ods have been widely used in statistics, probability, machine learning, and applied mathematics as well as many applications. Examples include low- rank matrix denoising (Shabalin and Nobel, 2013; Yang et al., 2014; Donoho and Gavish, 2014), matrix completion (Candès and Recht, 2009; Candès and Tao, 2010; Keshavan et al., 2010; Gross, 2011; Chatterjee, 2014), principle component analysis (Anderson, 2003; Johnstone and Lu, 2009; Cai et al., 2013, 2015b), canonical correlation analysis (Hotelling, 1936; Hardoon et al., 2004; Gao et al., 2014, 2015), community detection (von Luxburg et al., 2008; Rohe et al., 2011; Balakrishnan et al., 2011; Lei and Rinaldo, 2015). Specific applications include collaborative filtering (the Netflix problem) (Goldberg et al., 1992), multi-task learning (Argyriou et al., 2008), system identifi- cation (Liu and Vandenberghe, 2009), and sensor localization (Singer and

∗The research of Tony Cai was supported in part by NSF Grant DMS-1208982 and DMS-1403708, and NIH Grant R01 CA127334.

MSC 2010 subject classifications: Primary 62H12, 62C20; secondary 62H25 Keywords and phrases: canonical correlation analysis, clustering, high-dimensional

statistics, low-rank matrix denoising, perturbation bound, singular value decomposition, sin Θ distances, spectral method

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2 T. T. CAI AND A. ZHANG

Cucuringu, 2010; Candes and Plan, 2010), among many others. In addition, the SVD is often used to find a “warm start” for more delicate iterative algorithms, see, e.g., Cai et al. (2016); Sun and Luo (2015).

Perturbation bounds, which concern how the spectrum changes after a small perturbation to a matrix, often play a critical role in the analysis of the SVD and spectral methods. To be more specific, for an approximately low-rank matrix X and a perturbation matrix Z, it is crucial in many appli- cations to understand how much the left or right singular spaces of X and X + Z differ from each other. This problem has been widely studied in the literature (Davis and Kahan, 1970; Wedin, 1972; Weyl, 1912; Stewart, 1991, 2006; Yu et al., 2015). Among these results, the sin Θ theorems, established by Davis and Kahan (1970) and Wedin (1972), have become fundamental tools and are commonly used in applications. While Davis and Kahan (1970) focused on eigenvectors of symmetric matrices, Wedin’s sin Θ theorem stud- ies the more general singular vectors for asymmetric matrices and provides a uniform perturbation bound for both the left and right singular spaces in terms of the singular value gap and perturbation level.

Several generalizations and extensions have been made in different set- tings after the seminal work of Wedin (1972). For example, Vu (2011), Sha- balin and Nobel (2013), O’Rourke et al. (2013), Wang (2015) considered the rotations of singular vectors after random perturbations; Fan et al. (2016) gave an `∞ eigenvector perturbation bound and used the result for robust covariance estimation. See also Dopico (2000); Stewart (2006).

Despite its wide applicability, Wedin’s perturbation bound is not suffi- ciently precise for some analyses, as the bound is uniform for both the left and right singular spaces. It clearly leads to sub-optimal result if the left and right singular spaces change in different orders of magnitude after the perturbation. In a range of applications, especially when the row and col- umn dimensions of the matrix differ significantly, it is even possible that one side of the singular space can be accurately recovered, while the other side cannot. The numerical experiment given in Section 2.3 provides a good illustration for this point. It can be seen from the experiment that the left and right singular perturbation bounds behave distinctly when the row and column dimensions are significantly different. Furthermore, for a range of applications, the primary interest only lies in one of the singular spaces. For example, in the analysis of bipartite network data, such as the Face- book user-public-page-subscription network, the interest is often focused on grouping the public pages (or grouping the users). This is the case for many clustering problems. See Section 4 for further discussions.

In this paper, we establish separate perturbation bounds for the left and

PERTURBATION BOUNDS FOR SINGULAR SUBSPACES 3

right singular subspaces. The bounds are measured in both the spectral and Frobenius sin Θ distances, which are equivalent to several widely used losses in the literature. We also derive lower bounds that are within a constant factor of the corresponding upper bounds. These results together show that the obtained perturbation bounds are rate-optimal.

The newly established perturbation bounds are applicable to a wide range of problems in high-dimensional statistics. In this paper, we discuss in detail the applications of the perturbation bounds to the following high-dimensional statistical problems.

1. Low-rank matrix denoising and singular space estimation: Suppose one observes a low-rank matrix with random additive noise and wishes to estimate the mean matrix or its left or right singular spaces. Such a problem arises in many applications. We apply the obtained pertur- bation bounds to study this problem. Separate matching upper and lower bounds are given for estimating the left and right singular spaces. These results together establish the optimal rates of convergence. Our analysis shows an interesting phenomenon that in some settings it is possible to accurately estimate the left singular space but not the right one and vice versa. To the best of our knowledge, this is the first re- sult that gives different optimal rates for the left and right singular spaces under the same perturbation. Another fact we observe is that in certain class of low-rank matrices, one can stably recover the orig- inal matrix if and only if one can accurately recover both its left and right singular spaces.

2. High-dimensional clustering: Unsupervised learning is an important problem in statistics and machine learning with a wide range of ap- plications. We apply the perturbation bounds to the analysis of clus- tering for high-dimensional Gaussian mixtures. Particularly in a high- dimensional two-class clustering setting, we propose a simple PCA- based clustering method and use the obtained perturbation bounds to prove matching upper and lower bounds for the misclassification rates.

3. Canonical correlation analysis (CCA): CCA is a commonly used tools in multivariate analysis to identify and measure the associations among two sets of random variables. The perturbation bounds are also ap- plied to analyze CCA. Specifically, we develop sharper upper bounds for estimating the left and right canonical correlation directions. To the best of our knowledge, this is the first result that captures the phenomenon that in some settings it is possible to accurately estimate one side of canonical correlation directions but not the other side.

4 T. T. CAI AND A. ZHANG

In addition to these applications, the perturbation bounds can also be ap- plied to the analysis of community detection in bipartite networks, multi- dimensional scaling, cross-covariance matrix estimation, and singular space estimation for matrix completion, and other problems to yield better results than what are known in the literature. These applications demonstrate the usefulness of the newly established perturbation bounds.

The rest of the paper is organized as follows. In Section 2, after basic no- tation and definitions are introduced, the perturbation bounds are presented separately for the left and right singular subspaces. Both the upper bounds and lower bounds are provided. We then apply the newly established pertur- bation bounds to low-rank matrix denoising and singular space estimation, high-dimensional clustering, and canonical correlation analysis in Sections 3–5. Section 6 presents some numerical results and other potential applica- tions are briefly discussed in Section 7. The main theorems are proved in Section 8 and the proofs of some additional technical results are given in the supplementary materials.

2. Rate-Optim

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