In this paper, we resolve many of the key algorithmic questions regarding robustness, memory efficiency, and differential privacy of tensor decomposition. We propose simple variants of the tensor power method which enjoy these strong properties. We present the first guarantees for online tensor power method which has a linear memory requirement. Moreover, we present a noise calibrated tensor power method with efficient privacy guarantees. At the heart of all these guarantees lies a careful perturbation analysis derived in this paper which improves up on the existing results significantly.
Our relaxed noise condition analysis for tensor power method is inspired by recent analysis of noisy matrix power methods @cite_8 @cite_18 . Unlike the matrix case, tensor decomposition no longer requires among eigenvalues and eigenvectors are usually recovered one at a time @cite_23 @cite_13 . This poses new challenges and requires non-trivial extensions of matrix power method analysis to the tensor case.
@cite_18 @cite_13 @cite_23 @cite_8
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We consider the noisy power method algorithm, which has wide applications in machine learning and statistics, especially those related to principal component analysis (PCA) under resource (communication, memory or privacy) constraints. Existing analysis of the noisy power method shows an unsatisfactory dependency over the "consecutive" spectral gap @math of an input data matrix, which could be very small and hence limits the algorithm's applicability. In this paper, we present a new analysis of the noisy power method that achieves improved gap dependency for both sample complexity and noise tolerance bounds. More specifically, we improve the dependency over @math to dependency over @math , where @math is an intermediate algorithm parameter and could be much larger than the target rank @math . Our proofs are built upon a novel characterization of proximity between two subspaces that differ from canonical angle characterizations analyzed in previous works. Finally, we apply our improved bounds to distributed private PCA and memory-efficient streaming PCA and obtain bounds that are superior to existing results in the literature. We provide guarantees for learning latent variable models emphasizing on the overcomplete regime, where the dimensionality of the latent space exceeds the observed dimensionality. In particular, we consider multiview mixtures, ICA, and sparse coding models. Our main tool is a new algorithm for tensor decomposition that works in the overcomplete regime. In the semi-supervised setting, we exploit label information to get a rough estimate of the model parameters, and then refine it using the tensor method on unlabeled samples. We establish learning guarantees when the number of components scales as k = o(d), where d is the observed dimension, and p is the order of the observed moment employed in the tensor method (usually p = 3, 4). In the unsupervised setting, a simple initialization algorithm based on SVD of the tensor slices is proposed, and the guarantees are provided under the stricter condition that k ≤ βd (where constant β can be larger than 1). For the learning applications, we provide tight sample complexity bounds through novel covering arguments. This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models--including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation--which exploits a certain tensor structure in their low-order observable moments (typically, of second- and third-order). Specifically, parameter estimation is reduced to the problem of extracting a certain (orthogonal) decomposition of a symmetric tensor derived from the moments; this decomposition can be viewed as a natural generalization of the singular value decomposition for matrices. Although tensor decompositions are generally intractable to compute, the decomposition of these specially structured tensors can be efficiently obtained by a variety of approaches, including power iterations and maximization approaches (similar to the case of matrices). A detailed analysis of a robust tensor power method is provided, establishing an analogue of Wedin's perturbation theorem for the singular vectors of matrices. This implies a robust and computationally tractable estimation approach for several popular latent variable models. We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call the noisy power method. Our result characterizes the convergence behavior of the algorithm when a significant amount noise is introduced after each matrix-vector multiplication. The noisy power method can be seen as a meta-algorithm that has recently found a number of important applications in a broad range of machine learning problems including alternating minimization for matrix completion, streaming principal component analysis (PCA), and privacy-preserving spectral analysis. Our general analysis subsumes several existing ad-hoc convergence bounds and resolves a number of open problems in multiple applications including streaming PCA and privacy-preserving singular vector computation.