115 research outputs found
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
Overview of Constrained PARAFAC Models
In this paper, we present an overview of constrained PARAFAC models where the
constraints model linear dependencies among columns of the factor matrices of
the tensor decomposition, or alternatively, the pattern of interactions between
different modes of the tensor which are captured by the equivalent core tensor.
Some tensor prerequisites with a particular emphasis on mode combination using
Kronecker products of canonical vectors that makes easier matricization
operations, are first introduced. This Kronecker product based approach is also
formulated in terms of the index notation, which provides an original and
concise formalism for both matricizing tensors and writing tensor models. Then,
after a brief reminder of PARAFAC and Tucker models, two families of
constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models,
are described in a unified framework, for order tensors. New tensor
models, called nested Tucker models and block PARALIND/CONFAC models, are also
introduced. A link between PARATUCK models and constrained PARAFAC models is
then established. Finally, new uniqueness properties of PARATUCK models are
deduced from sufficient conditions for essential uniqueness of their associated
constrained PARAFAC models
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
Multimodal Data Fusion: An Overview of Methods, Challenges and Prospects
International audienceIn various disciplines, information about the same phenomenon can be acquired from different types of detectors, at different conditions, in multiple experiments or subjects, among others. We use the term "modality" for each such acquisition framework. Due to the rich characteristics of natural phenomena, it is rare that a single modality provides complete knowledge of the phenomenon of interest. The increasing availability of several modalities reporting on the same system introduces new degrees of freedom, which raise questions beyond those related to exploiting each modality separately. As we argue, many of these questions, or "challenges" , are common to multiple domains. This paper deals with two key questions: "why we need data fusion" and "how we perform it". The first question is motivated by numerous examples in science and technology, followed by a mathematical framework that showcases some of the benefits that data fusion provides. In order to address the second question, "diversity" is introduced as a key concept, and a number of data-driven solutions based on matrix and tensor decompositions are discussed, emphasizing how they account for diversity across the datasets. The aim of this paper is to provide the reader, regardless of his or her community of origin, with a taste of the vastness of the field, the prospects and opportunities that it holds
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
Approximate Rank-Detecting Factorization of Low-Rank Tensors
We present an algorithm, AROFAC2, which detects the (CP-)rank of a degree 3
tensor and calculates its factorization into rank-one components. We provide
generative conditions for the algorithm to work and demonstrate on both
synthetic and real world data that AROFAC2 is a potentially outperforming
alternative to the gold standard PARAFAC over which it has the advantages that
it can intrinsically detect the true rank, avoids spurious components, and is
stable with respect to outliers and non-Gaussian noise
- …