394 research outputs found
On the typical rank of real binary forms
We determine the rank of a general real binary form of degree d=4 and d=5. In
the case d=5, the possible values of the rank of such general forms are 3,4,5.
The existence of three typical ranks was unexpected. We prove that a real
binary form of degree d with d real roots has rank d.Comment: 12 pages, 2 figure
Symmetric tensor decomposition
We present an algorithm for decomposing a symmetric tensor, of dimension n
and order d as a sum of rank-1 symmetric tensors, extending the algorithm of
Sylvester devised in 1886 for binary forms. We recall the correspondence
between the decomposition of a homogeneous polynomial in n variables of total
degree d as a sum of powers of linear forms (Waring's problem), incidence
properties on secant varieties of the Veronese Variety and the representation
of linear forms as a linear combination of evaluations at distinct points. Then
we reformulate Sylvester's approach from the dual point of view. Exploiting
this duality, we propose necessary and sufficient conditions for the existence
of such a decomposition of a given rank, using the properties of Hankel (and
quasi-Hankel) matrices, derived from multivariate polynomials and normal form
computations. This leads to the resolution of polynomial equations of small
degree in non-generic cases. We propose a new algorithm for symmetric tensor
decomposition, based on this characterization and on linear algebra
computations with these Hankel matrices. The impact of this contribution is
two-fold. First it permits an efficient computation of the decomposition of any
tensor of sub-generic rank, as opposed to widely used iterative algorithms with
unproved global convergence (e.g. Alternate Least Squares or gradient
descents). Second, it gives tools for understanding uniqueness conditions, and
for detecting the rank
Nonnegative approximations of nonnegative tensors
We study the decomposition of a nonnegative tensor into a minimal sum of
outer product of nonnegative vectors and the associated parsimonious naive
Bayes probabilistic model. We show that the corresponding approximation
problem, which is central to nonnegative PARAFAC, will always have optimal
solutions. The result holds for any choice of norms and, under a mild
assumption, even Bregman divergences.Comment: 14 page
Blind Multilinear Identification
We discuss a technique that allows blind recovery of signals or blind
identification of mixtures in instances where such recovery or identification
were previously thought to be impossible: (i) closely located or highly
correlated sources in antenna array processing, (ii) highly correlated
spreading codes in CDMA radio communication, (iii) nearly dependent spectra in
fluorescent spectroscopy. This has important implications --- in the case of
antenna array processing, it allows for joint localization and extraction of
multiple sources from the measurement of a noisy mixture recorded on multiple
sensors in an entirely deterministic manner. In the case of CDMA, it allows the
possibility of having a number of users larger than the spreading gain. In the
case of fluorescent spectroscopy, it allows for detection of nearly identical
chemical constituents. The proposed technique involves the solution of a
bounded coherence low-rank multilinear approximation problem. We show that
bounded coherence allows us to establish existence and uniqueness of the
recovered solution. We will provide some statistical motivation for the
approximation problem and discuss greedy approximation bounds. To provide the
theoretical underpinnings for this technique, we develop a corresponding theory
of sparse separable decompositions of functions, including notions of rank and
nuclear norm that specialize to the usual ones for matrices and operators but
apply to also hypermatrices and tensors.Comment: 20 pages, to appear in IEEE Transactions on Information Theor
Multiarray Signal Processing: Tensor decomposition meets compressed sensing
We discuss how recently discovered techniques and tools from compressed
sensing can be used in tensor decompositions, with a view towards modeling
signals from multiple arrays of multiple sensors. We show that with appropriate
bounds on a measure of separation between radiating sources called coherence,
one could always guarantee the existence and uniqueness of a best rank-r
approximation of the tensor representing the signal. We also deduce a
computationally feasible variant of Kruskal's uniqueness condition, where the
coherence appears as a proxy for k-rank. Problems of sparsest recovery with an
infinite continuous dictionary, lowest-rank tensor representation, and blind
source separation are treated in a uniform fashion. The decomposition of the
measurement tensor leads to simultaneous localization and extraction of
radiating sources, in an entirely deterministic manner.Comment: 10 pages, 1 figur
Approximate matrix and tensor diagonalization by unitary transformations: convergence of Jacobi-type algorithms
We propose a gradient-based Jacobi algorithm for a class of maximization
problems on the unitary group, with a focus on approximate diagonalization of
complex matrices and tensors by unitary transformations. We provide weak
convergence results, and prove local linear convergence of this algorithm.The
convergence results also apply to the case of real-valued tensors
Tensors: a Brief Introduction
International audienceTensor decompositions are at the core of many Blind Source Separation (BSS) algorithms, either explicitly or implicitly. In particular, the Canonical Polyadic (CP) tensor decomposition plays a central role in identification of underdetermined mixtures. Despite some similarities, CP and Singular value Decomposition (SVD) are quite different. More generally, tensors and matrices enjoy different properties, as pointed out in this brief survey
On the minimal ranks of matrix pencils and the existence of a best approximate block-term tensor decomposition
Under the action of the general linear group with tensor structure, the ranks
of matrices and forming an pencil can
change, but in a restricted manner. Specifically, with every pencil one can
associate a pair of minimal ranks, which is unique up to a permutation. This
notion can be defined for matrix pencils and, more generally, also for matrix
polynomials of arbitrary degree. In this paper, we provide a formal definition
of the minimal ranks, discuss its properties and the natural hierarchy it
induces in a pencil space. Then, we show how the minimal ranks of a pencil can
be determined from its Kronecker canonical form. For illustration, we classify
the orbits according to their minimal ranks (under the action of the general
linear group) in the case of real pencils with . Subsequently, we
show that real regular pencils having only complex-valued
eigenvalues, which form an open positive-volume set, do not admit a best
approximation (in the norm topology) on the set of real pencils whose minimal
ranks are bounded by . Our results can be interpreted from a tensor
viewpoint, where the minimal ranks of a degree- matrix polynomial
characterize the minimal ranks of matrices constituting a block-term
decomposition of an tensor into a sum of matrix-vector
tensor products.Comment: This work was supported by the European Research Council under the
European Programme FP7/2007-2013, Grant AdG-2013-320594 "DECODA.
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