3,715 research outputs found
Rank-1 Tensor Approximation Methods and Application to Deflation
Because of the attractiveness of the canonical polyadic (CP) tensor
decomposition in various applications, several algorithms have been designed to
compute it, but efficient ones are still lacking. Iterative deflation
algorithms based on successive rank-1 approximations can be used to perform
this task, since the latter are rather easy to compute. We first present an
algebraic rank-1 approximation method that performs better than the standard
higher-order singular value decomposition (HOSVD) for three-way tensors.
Second, we propose a new iterative rank-1 approximation algorithm that improves
any other rank-1 approximation method. Third, we describe a probabilistic
framework allowing to study the convergence of deflation CP decomposition
(DCPD) algorithms based on successive rank-1 approximations. A set of computer
experiments then validates theoretical results and demonstrates the efficiency
of DCPD algorithms compared to other ones
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
Uniqueness of Nonnegative Tensor Approximations
We show that for a nonnegative tensor, a best nonnegative rank-r
approximation is almost always unique, its best rank-one approximation may
always be chosen to be a best nonnegative rank-one approximation, and that the
set of nonnegative tensors with non-unique best rank-one approximations form an
algebraic hypersurface. We show that the last part holds true more generally
for real tensors and thereby determine a polynomial equation so that a real or
nonnegative tensor which does not satisfy this equation is guaranteed to have a
unique best rank-one approximation. We also establish an analogue for real or
nonnegative symmetric tensors. In addition, we prove a singular vector variant
of the Perron--Frobenius Theorem for positive tensors and apply it to show that
a best nonnegative rank-r approximation of a positive tensor can never be
obtained by deflation. As an aside, we verify that the Euclidean distance (ED)
discriminants of the Segre variety and the Veronese variety are hypersurfaces
and give defining equations of these ED discriminants
Subtracting a best rank-1 approximation may increase tensor rank
It has been shown that a best rank-R approximation of an order-k tensor may
not exist when R>1 and k>2. This poses a serious problem to data analysts using
tensor decompositions. It has been observed numerically that, generally, this
issue cannot be solved by consecutively computing and subtracting best rank-1
approximations. The reason for this is that subtracting a best rank-1
approximation generally does not decrease tensor rank. In this paper, we
provide a mathematical treatment of this property for real-valued 2x2x2
tensors, with symmetric tensors as a special case. Regardless of the symmetry,
we show that for generic 2x2x2 tensors (which have rank 2 or 3), subtracting a
best rank-1 approximation results in a tensor that has rank 3 and lies on the
boundary between the rank-2 and rank-3 sets. Hence, for a typical tensor of
rank 2, subtracting a best rank-1 approximation increases the tensor rank.Comment: 37 page
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
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