Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOrthogonal Decomposition of Tensor Trains
123
0
0.0
(
0
)
Added by
Elina Robeva
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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 propose a new algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this as a system of polynomial equations allows us to leverage recent numerical linear algebra tools from computational algebraic geometry. We describe the complexity of our algorithm in terms of the multigraded regularity of a multihomogeneous ideal. We prove effective bounds for many formats and ranks and conjecture a general formula. These bounds are of independent interest for overconstrained polynomial system solving. Our experiments show that our algorithm can outperform state-of-the-art algebraic algorithms by an order of magnitude in terms of accuracy, computation time, and memory consumption.
Differential algebraic Riccati equations are at the heart of many applications in control theory. They are time-depent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been used heavily computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton-Kleinman iteration. Approximating the space-time problem in low-rank form requires fewer applications of the discretized differential operator and gives a low-rank approximation to the overall solution.
In this paper, we consider the tensor completion problem, which has many researchers in the machine learning particularly concerned. Our fast and precise method is built on extending the $L_{2,1}$-norm minimization and Qatar Riyal decomposition (LNM-QR) method for matrix completions to tensor completions, and is different from the popular tensor completion methods using the tensor singular value decomposition (t-SVD). In terms of shortening the computing time, t-SVD is replaced with the method computing an approximate t-SVD based on Qatar Riyal decomposition (CTSVD-QR), which can be used to compute the largest $r left(r>0 right)$ singular values (tubes) and their associated singular vectors (of tubes) iteratively. We, in addition, use the tensor $L_{2,1}$-norm instead of the tensor nuclear norm to minimize our model on account of it is easy to optimize. Then in terms of improving accuracy, ADMM, a gradient-search-based method, plays a crucial part in our method. Numerical experimental results show that our method is faster than those state-of-the-art algorithms and have excellent accuracy.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
