Randomized Dimension Reduction on Massive Data

Scalability of statistical estimators is of increasing importance in modern applications and dimension reduction is often used to extract relevant information from data. A variety of popular dimension reduction approaches can be framed as symmetric generalized eigendecomposition problems. In this paper we outline how taking into account the low rank structure assumption implicit in these dimension reduction approaches provides both computational and statistical advantages. We adapt recent randomized low-rank approximation algorithms to provide efficient solutions to three dimension reduction methods: Principal Component Analysis (PCA), Sliced Inverse Regression (SIR), and Localized Sliced Inverse Regression (LSIR). A key observation in this paper is that randomization serves a dual role, improving both computational and statistical performance. This point is highlighted in our experiments on real and simulated data.Comment: 31 pages, 6 figures, Key Words:dimension reduction, generalized eigendecompositon, low-rank, supervised, inverse regression, random projections, randomized algorithms, Krylov subspace method

Adaptive Higher-order Spectral Estimators

Many applications involve estimation of a signal matrix from a noisy data matrix. In such cases, it has been observed that estimators that shrink or truncate the singular values of the data matrix perform well when the signal matrix has approximately low rank. In this article, we generalize this approach to the estimation of a tensor of parameters from noisy tensor data. We develop new classes of estimators that shrink or threshold the mode-specific singular values from the higher-order singular value decomposition. These classes of estimators are indexed by tuning parameters, which we adaptively choose from the data by minimizing Stein's unbiased risk estimate. In particular, this procedure provides a way to estimate the multilinear rank of the underlying signal tensor. Using simulation studies under a variety of conditions, we show that our estimators perform well when the mean tensor has approximately low multilinear rank, and perform competitively when the signal tensor does not have approximately low multilinear rank. We illustrate the use of these methods in an application to multivariate relational data.Comment: 29 pages, 3 figure