Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSymmetric Orthogonal Tensor Decomposition is Trivial
220
0
0.0
(
0
)
Added by
Tamara Kolda
Publication date
2015
fields
Informatics Engineering
and research's language is
English
Authors
Tamara G. Kolda
Ask ChatGPT about the research
No Arabic abstract
We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued, pairwise orthogonal vectors. Such decompositions do not generally exist, but we show that some symmetric tensor decomposition problems can be converted to orthogonal problems following the whitening procedure proposed by Anandkumar et al. (2012). If an orthogonal decomposition of an $m$-way $n$-dimensional symmetric tensor exists, we propose a novel method to compute it that reduces to an $n times n$ symmetric matrix eigenproblem. We provide numerical results demonstrating the effectiveness of the method.
rate research
Read More
In this paper we study the problem of recovering a tensor network decomposition of a given tensor $mathcal{T}$ in which the tensors at the vertices of the network are orthogonally decomposable. Specifically, we consider tensor networks in the form of tensor trains (aka matrix product states). When the tensor train has length 2, and the orthogonally decomposable tensors at the two vertices of the network are symmetric, we show how to recover the decomposition by considering random linear combinations of slices. Furthermore, if the tensors at the vertices are symmetric but not orthogonally decomposable, we show that a whitening procedure can transform the problem into an orthogonal one, thereby yielding a solution for the decomposition of the tensor. When the tensor network has length 3 or more and the tensors at the vertices are symmetric and orthogonally decomposable, we provide an algorithm for recovering them subject to some rank conditions. Finally, in the case of tensor trains of length two in which the tensors at the vertices are orthogonally decomposable but not necessarily symmetric, we show that the decomposition problem reduces to the problem of a novel matrix decomposition, that of an orthogonal matrix multiplied by diagonal matrices on either side. We provide two solutions for the full-rank tensor case using Sinkhorns theorem and Procrustes algorithm, respectively, and show that the Procrustes-based solution can be generalized to any rank case. We conclude with a multitude of open problems in linear and multilinear algebra that arose in our study.
We consider the problem of decomposing higher-order moment tensors, i.e., the sum of symmetric outer products of data vectors. Such a decomposition can be used to estimate the means in a Gaussian mixture model and for other applications in machine learning. The $d$th-order empirical moment tensor of a set of $p$ observations of $n$ variables is a symmetric $d$-way tensor. Our goal is to find a low-rank tensor approximation comprising $r ll p$ symmetric outer products. The challenge is that forming the empirical moment tensors costs $O(pn^d)$ operations and $O(n^d)$ storage, which may be prohibitively expensive; additionally, the algorithm to compute the low-rank approximation costs $O(n^d)$ per iteration. Our contribution is avoiding formation of the moment tensor, computing the low-rank tensor approximation of the moment tensor implicitly using $O(pnr)$ operations per iteration and no extra memory. This advance opens the door to more applications of higher-order moments since they can now be efficiently computed. We present numerical evidence of the computational savings and show an example of estimating the means for higher-order moments.
We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank even-order real symmetric tensors. This algorithm applies the tensor power method of Kolda-Mayo to a certain modified tensor, constructed from a matrix flattening of the original tensor, and then uses deflation steps. Numerical simulations indicate SPM is roughly one order of magnitude faster than state-of-the-art algorithms, while performing robustly for low-rank tensors subjected to additive noise. We obtain rigorous guarantees for SPM regarding convergence and global optima, for tensors of rank up to roughly the square root of the number of tensor entries, by drawing on results from classical algebraic geometry and dynamical systems. In a second contribution, we extend SPM to compute De Lathauwers symmetric block term tensor decompositions. As an application of the latter decomposition, we provide a method-of-moments for generalized principal component analysis.
We study the problem of finding orthogonal low-rank approximations of symmetric tensors. In the case of matrices, the approximation is a truncated singular value decomposition which is then symmetric. Moreover, for rank-one approximations of tensors of any dimension, a classical result proven by Banach in 1938 shows that the optimal approximation can always be chosen to be symmetric. In contrast to these results, this article shows that the corresponding statement is no longer true for orthogonal approximations of higher rank. Specifically, for any of the four common notions of tensor orthogonality used in the literature, we show that optimal orthogonal approximations of rank greater than one cannot always be chosen to be symmetric.
This paper discusses the problem of symmetric tensor decomposition on a given variety $X$: decomposing a symmetric tensor into the sum of tensor powers of vectors contained in $X$. In this paper, we first study geometric and algebraic properties of such decomposable tensors, which are crucial to the practical computations of such decompositions. For a given tensor, we also develop a criterion for the existence of a symmetric decomposition on $X$. Secondly and most importantly, we propose a method for computing symmetric tensor decompositions on an arbitrary $X$. As a specific application, Vandermonde decompositions for nonsymmetric tensors can be computed by the proposed algorithm.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
