2,471 research outputs found
Fast Decomposition of Large Nonnegative Tensors
International audienceIn Signal processing, tensor decompositions have gained in popularity this last decade. In the meantime, the volume of data to be processed has drastically increased. This calls for novel methods to handle Big Data tensors. Since most of these huge data are issued from physical measurements, which are intrinsically real nonnegative, being able to compress nonnegative tensors has become mandatory. Following recent works on HOSVD compression for Big Data, we detail solutions to decompose a nonnegative tensor into decomposable terms in a compressed domain
Dictionary-based Tensor Canonical Polyadic Decomposition
To ensure interpretability of extracted sources in tensor decomposition, we
introduce in this paper a dictionary-based tensor canonical polyadic
decomposition which enforces one factor to belong exactly to a known
dictionary. A new formulation of sparse coding is proposed which enables high
dimensional tensors dictionary-based canonical polyadic decomposition. The
benefits of using a dictionary in tensor decomposition models are explored both
in terms of parameter identifiability and estimation accuracy. Performances of
the proposed algorithms are evaluated on the decomposition of simulated data
and the unmixing of hyperspectral images
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
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