884 research outputs found
A constructive arbitrary-degree Kronecker product decomposition of tensors
We propose the tensor Kronecker product singular value decomposition~(TKPSVD)
that decomposes a real -way tensor into a linear combination
of tensor Kronecker products with an arbitrary number of factors
. We generalize the matrix Kronecker product to
tensors such that each factor in the TKPSVD is a -way
tensor. The algorithm relies on reshaping and permuting the original tensor
into a -way tensor, after which a polyadic decomposition with orthogonal
rank-1 terms is computed. We prove that for many different structured tensors,
the Kronecker product factors
are guaranteed to inherit this structure. In addition, we introduce the new
notion of general symmetric tensors, which includes many different structures
such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors.Comment: Rewrote the paper completely and generalized everything to tensor
Symmetric Tensor Decomposition by an Iterative Eigendecomposition Algorithm
We present an iterative algorithm, called the symmetric tensor eigen-rank-one
iterative decomposition (STEROID), for decomposing a symmetric tensor into a
real linear combination of symmetric rank-1 unit-norm outer factors using only
eigendecompositions and least-squares fitting. Originally designed for a
symmetric tensor with an order being a power of two, STEROID is shown to be
applicable to any order through an innovative tensor embedding technique.
Numerical examples demonstrate the high efficiency and accuracy of the proposed
scheme even for large scale problems. Furthermore, we show how STEROID readily
solves a problem in nonlinear block-structured system identification and
nonlinear state-space identification
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
Low-rank approximate inverse for preconditioning tensor-structured linear systems
In this paper, we propose an algorithm for the construction of low-rank
approximations of the inverse of an operator given in low-rank tensor format.
The construction relies on an updated greedy algorithm for the minimization of
a suitable distance to the inverse operator. It provides a sequence of
approximations that are defined as the projections of the inverse operator in
an increasing sequence of linear subspaces of operators. These subspaces are
obtained by the tensorization of bases of operators that are constructed from
successive rank-one corrections. In order to handle high-order tensors,
approximate projections are computed in low-rank Hierarchical Tucker subsets of
the successive subspaces of operators. Some desired properties such as symmetry
or sparsity can be imposed on the approximate inverse operator during the
correction step, where an optimal rank-one correction is searched as the tensor
product of operators with the desired properties. Numerical examples illustrate
the ability of this algorithm to provide efficient preconditioners for linear
systems in tensor format that improve the convergence of iterative solvers and
also the quality of the resulting low-rank approximations of the solution
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