368 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
Joint Sensing Matrix and Sparsifying Dictionary Optimization for Tensor Compressive Sensing.
Tensor compressive sensing (TCS) is a multidimensional framework of compressive sensing (CS), and it is
advantageous in terms of reducing the amount of storage, easing
hardware implementations, and preserving multidimensional
structures of signals in comparison to a conventional CS system.
In a TCS system, instead of using a random sensing matrix and
a predefined dictionary, the average-case performance can be
further improved by employing an optimized multidimensional
sensing matrix and a learned multilinear sparsifying dictionary.
In this paper, we propose an approach that jointly optimizes
the sensing matrix and dictionary for a TCS system. For the
sensing matrix design in TCS, an extended separable approach
with a closed form solution and a novel iterative nonseparable
method are proposed when the multilinear dictionary is fixed.
In addition, a multidimensional dictionary learning method that
takes advantages of the multidimensional structure is derived,
and the influence of sensing matrices is taken into account in the
learning process. A joint optimization is achieved via alternately
iterating the optimization of the sensing matrix and dictionary.
Numerical experiments using both synthetic data and real images
demonstrate the superiority of the proposed approache
Joint Tensor Factorization and Outlying Slab Suppression with Applications
We consider factoring low-rank tensors in the presence of outlying slabs.
This problem is important in practice, because data collected in many
real-world applications, such as speech, fluorescence, and some social network
data, fit this paradigm. Prior work tackles this problem by iteratively
selecting a fixed number of slabs and fitting, a procedure which may not
converge. We formulate this problem from a group-sparsity promoting point of
view, and propose an alternating optimization framework to handle the
corresponding () minimization-based low-rank tensor
factorization problem. The proposed algorithm features a similar per-iteration
complexity as the plain trilinear alternating least squares (TALS) algorithm.
Convergence of the proposed algorithm is also easy to analyze under the
framework of alternating optimization and its variants. In addition,
regularization and constraints can be easily incorporated to make use of
\emph{a priori} information on the latent loading factors. Simulations and real
data experiments on blind speech separation, fluorescence data analysis, and
social network mining are used to showcase the effectiveness of the proposed
algorithm
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