General Tensor Decomposition, Moment Matrices and Applications

SubmittedInternational audienceThe tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation

Training Input-Output Recurrent Neural Networks through Spectral Methods

We consider the problem of training input-output recurrent neural networks (RNN) for sequence labeling tasks. We propose a novel spectral approach for learning the network parameters. It is based on decomposition of the cross-moment tensor between the output and a non-linear transformation of the input, based on score functions. We guarantee consistent learning with polynomial sample and computational complexity under transparent conditions such as non-degeneracy of model parameters, polynomial activations for the neurons, and a Markovian evolution of the input sequence. We also extend our results to Bidirectional RNN which uses both previous and future information to output the label at each time point, and is employed in many NLP tasks such as POS tagging

Sample Complexity Analysis for Learning Overcomplete Latent Variable Models through Tensor Methods

We provide guarantees for learning latent variable models emphasizing on the overcomplete regime, where the dimensionality of the latent space can exceed the observed dimensionality. In particular, we consider multiview mixtures, spherical Gaussian mixtures, ICA, and sparse coding models. We provide tight concentration bounds for empirical moments through novel covering arguments. We analyze parameter recovery through a simple tensor power update algorithm. In the semi-supervised setting, we exploit the label or prior information to get a rough estimate of the model parameters, and then refine it using the tensor method on unlabeled samples. We establish that learning is possible when the number of components scales as $k=o(d^{p/2})$, where $d$ is the observed dimension, and $p$ is the order of the observed moment employed in the tensor method. Our concentration bound analysis also leads to minimax sample complexity for semi-supervised learning of spherical Gaussian mixtures. In the unsupervised setting, we use a simple initialization algorithm based on SVD of the tensor slices, and provide guarantees under the stricter condition that $k\le \beta d$ (where constant $\beta$ can be larger than $1$), where the tensor method recovers the components under a polynomial running time (and exponential in $\beta$). Our analysis establishes that a wide range of overcomplete latent variable models can be learned efficiently with low computational and sample complexity through tensor decomposition methods.Comment: Title change

Symmetric tensor decomposition

We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring's problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester's approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the rank

Fourier PCA and Robust Tensor Decomposition

Fourier PCA is Principal Component Analysis of a matrix obtained from higher order derivatives of the logarithm of the Fourier transform of a distribution.We make this method algorithmic by developing a tensor decomposition method for a pair of tensors sharing the same vectors in rank-$1$ decompositions. Our main application is the first provably polynomial-time algorithm for underdetermined ICA, i.e., learning an $n \times m$ matrix $A$ from observations $y=Ax$ where $x$ is drawn from an unknown product distribution with arbitrary non-Gaussian components. The number of component distributions $m$ can be arbitrarily higher than the dimension $n$ and the columns of $A$ only need to satisfy a natural and efficiently verifiable nondegeneracy condition. As a second application, we give an alternative algorithm for learning mixtures of spherical Gaussians with linearly independent means. These results also hold in the presence of Gaussian noise.Comment: Extensively revised; details added; minor errors corrected; exposition improve

Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition

Hankel tensors arise from applications such as signal processing. In this paper, we make an initial study on Hankel tensors. For each Hankel tensor, we associate it with a Hankel matrix and a higher order two-dimensional symmetric tensor, which we call the associated plane tensor. If the associated Hankel matrix is positive semi-definite, we call such a Hankel tensor a strong Hankel tensor. We show that an $m$ order $n$-dimensional tensor is a Hankel tensor if and only if it has a Vandermonde decomposition. We call a Hankel tensor a complete Hankel tensor if it has a Vandermonde decomposition with positive coefficients. We prove that if a Hankel tensor is copositive or an even order Hankel tensor is positive semi-definite, then the associated plane tensor is copositive or positive semi-definite, respectively. We show that even order strong and complete Hankel tensors are positive semi-definite, the Hadamard product of two strong Hankel tensors is a strong Hankel tensor, and the Hadamard product of two complete Hankel tensors is a complete Hankel tensor. We show that all the H-eigenvalue of a complete Hankel tensors (maybe of odd order) are nonnegative. We give some upper bounds and lower bounds for the smallest and the largest Z-eigenvalues of a Hankel tensor, respectively. Further questions on Hankel tensors are raised

Tensor decompositions for learning latent variable models

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