6,317 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
Tensor Networks for Big Data Analytics and Large-Scale Optimization Problems
In this paper we review basic and emerging models and associated algorithms
for large-scale tensor networks, especially Tensor Train (TT) decompositions
using novel mathematical and graphical representations. We discus the concept
of tensorization (i.e., creating very high-order tensors from lower-order
original data) and super compression of data achieved via quantized tensor
train (QTT) networks. The purpose of a tensorization and quantization is to
achieve, via low-rank tensor approximations "super" compression, and
meaningful, compact representation of structured data. The main objective of
this paper is to show how tensor networks can be used to solve a wide class of
big data optimization problems (that are far from tractable by classical
numerical methods) by applying tensorization and performing all operations
using relatively small size matrices and tensors and applying iteratively
optimized and approximative tensor contractions.
Keywords: Tensor networks, tensor train (TT) decompositions, matrix product
states (MPS), matrix product operators (MPO), basic tensor operations,
tensorization, distributed representation od data optimization problems for
very large-scale problems: generalized eigenvalue decomposition (GEVD),
PCA/SVD, canonical correlation analysis (CCA).Comment: arXiv admin note: text overlap with arXiv:1403.204
Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
The widespread use of multi-sensor technology and the emergence of big
datasets has highlighted the limitations of standard flat-view matrix models
and the necessity to move towards more versatile data analysis tools. We show
that higher-order tensors (i.e., multiway arrays) enable such a fundamental
paradigm shift towards models that are essentially polynomial and whose
uniqueness, unlike the matrix methods, is guaranteed under verymild and natural
conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical
backbone, data analysis techniques using tensor decompositions are shown to
have great flexibility in the choice of constraints that match data properties,
and to find more general latent components in the data than matrix-based
methods. A comprehensive introduction to tensor decompositions is provided from
a signal processing perspective, starting from the algebraic foundations, via
basic Canonical Polyadic and Tucker models, through to advanced cause-effect
and multi-view data analysis schemes. We show that tensor decompositions enable
natural generalizations of some commonly used signal processing paradigms, such
as canonical correlation and subspace techniques, signal separation, linear
regression, feature extraction and classification. We also cover computational
aspects, and point out how ideas from compressed sensing and scientific
computing may be used for addressing the otherwise unmanageable storage and
manipulation problems associated with big datasets. The concepts are supported
by illustrative real world case studies illuminating the benefits of the tensor
framework, as efficient and promising tools for modern signal processing, data
analysis and machine learning applications; these benefits also extend to
vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker
decomposition, HOSVD, tensor networks, Tensor Train
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