228 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
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
Knowledge Graph Completion via Complex Tensor Factorization
In statistical relational learning, knowledge graph completion deals with automatically
understanding the structure of large knowledge graphs—labeled directed graphs—and predicting missing relationships—labeled edges. State-of-the-art embedding models
propose different trade-offs between modeling expressiveness, and time and space complexity.
We reconcile both expressiveness and complexity through the use of complex-valued
embeddings and explore the link between such complex-valued embeddings and unitary
diagonalization. We corroborate our approach theoretically and show that all real square
matrices—thus all possible relation/adjacency matrices—are the real part of some unitarily
diagonalizable matrix. This results opens the door to a lot of other applications of square
matrices factorization. Our approach based on complex embeddings is arguably simple,
as it only involves a Hermitian dot product, the complex counterpart of the standard dot
product between real vectors, whereas other methods resort to more and more complicated
composition functions to increase their expressiveness. The proposed complex embeddings
are scalable to large data sets as it remains linear in both space and time, while consistently
outperforming alternative approaches on standard link prediction benchmarks
A condition number for the tensor rank decomposition
The tensor rank decomposition problem consists of recovering the unique set
of parameters representing a robustly identifiable low-rank tensor when the
coordinate representation of the tensor is presented as input. A condition
number for this problem measuring the sensitivity of the parameters to an
infinitesimal change to the tensor is introduced and analyzed. It is
demonstrated that the absolute condition number coincides with the inverse of
the least singular value of Terracini's matrix. Several basic properties of
this condition number are investigated.Comment: 45 pages, 4 figure
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