153 research outputs found
A Constructive Algorithm for Decomposing a Tensor into a Finite Sum of Orthonormal Rank-1 Terms
We propose a constructive algorithm that decomposes an arbitrary real tensor
into a finite sum of orthonormal rank-1 outer products. The algorithm, named
TTr1SVD, works by converting the tensor into a tensor-train rank-1 (TTr1)
series via the singular value decomposition (SVD). TTr1SVD naturally
generalizes the SVD to the tensor regime with properties such as uniqueness for
a fixed order of indices, orthogonal rank-1 outer product terms, and easy
truncation error quantification. Using an outer product column table it also
allows, for the first time, a complete characterization of all tensors
orthogonal with the original tensor. Incidentally, this leads to a strikingly
simple constructive proof showing that the maximum rank of a real tensor over the real field is 3. We also derive a conversion of the
TTr1 decomposition into a Tucker decomposition with a sparse core tensor.
Numerical examples illustrate each of the favorable properties of the TTr1
decomposition.Comment: Added subsection on orthogonal complement tensors. Added constructive
proof of maximal CP-rank of a 2x2x2 tensor. Added perturbation of singular
values result. Added conversion of the TTr1 decomposition to the Tucker
decomposition. Added example that demonstrates how the rank behaves when
subtracting rank-1 terms. Added example with exponential decaying singular
value
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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