223 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
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
A geometric Newton method for Oja's vector field
Newton's method for solving the matrix equation runs
up against the fact that its zeros are not isolated. This is due to a symmetry
of by the action of the orthogonal group. We show how
differential-geometric techniques can be exploited to remove this symmetry and
obtain a ``geometric'' Newton algorithm that finds the zeros of . The
geometric Newton method does not suffer from the degeneracy issue that stands
in the way of the original Newton method
Tensor Regression with Applications in Neuroimaging Data Analysis
Classical regression methods treat covariates as a vector and estimate a
corresponding vector of regression coefficients. Modern applications in medical
imaging generate covariates of more complex form such as multidimensional
arrays (tensors). Traditional statistical and computational methods are proving
insufficient for analysis of these high-throughput data due to their ultrahigh
dimensionality as well as complex structure. In this article, we propose a new
family of tensor regression models that efficiently exploit the special
structure of tensor covariates. Under this framework, ultrahigh dimensionality
is reduced to a manageable level, resulting in efficient estimation and
prediction. A fast and highly scalable estimation algorithm is proposed for
maximum likelihood estimation and its associated asymptotic properties are
studied. Effectiveness of the new methods is demonstrated on both synthetic and
real MRI imaging data.Comment: 27 pages, 4 figure
Tensor completion in hierarchical tensor representations
Compressed sensing extends from the recovery of sparse vectors from
undersampled measurements via efficient algorithms to the recovery of matrices
of low rank from incomplete information. Here we consider a further extension
to the reconstruction of tensors of low multi-linear rank in recently
introduced hierarchical tensor formats from a small number of measurements.
Hierarchical tensors are a flexible generalization of the well-known Tucker
representation, which have the advantage that the number of degrees of freedom
of a low rank tensor does not scale exponentially with the order of the tensor.
While corresponding tensor decompositions can be computed efficiently via
successive applications of (matrix) singular value decompositions, some
important properties of the singular value decomposition do not extend from the
matrix to the tensor case. This results in major computational and theoretical
difficulties in designing and analyzing algorithms for low rank tensor
recovery. For instance, a canonical analogue of the tensor nuclear norm is
NP-hard to compute in general, which is in stark contrast to the matrix case.
In this book chapter we consider versions of iterative hard thresholding
schemes adapted to hierarchical tensor formats. A variant builds on methods
from Riemannian optimization and uses a retraction mapping from the tangent
space of the manifold of low rank tensors back to this manifold. We provide
first partial convergence results based on a tensor version of the restricted
isometry property (TRIP) of the measurement map. Moreover, an estimate of the
number of measurements is provided that ensures the TRIP of a given tensor rank
with high probability for Gaussian measurement maps.Comment: revised version, to be published in Compressed Sensing and Its
Applications (edited by H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral
Tag-Aware Recommender Systems: A State-of-the-art Survey
In the past decade, Social Tagging Systems have attracted increasing
attention from both physical and computer science communities. Besides the
underlying structure and dynamics of tagging systems, many efforts have been
addressed to unify tagging information to reveal user behaviors and
preferences, extract the latent semantic relations among items, make
recommendations, and so on. Specifically, this article summarizes recent
progress about tag-aware recommender systems, emphasizing on the contributions
from three mainstream perspectives and approaches: network-based methods,
tensor-based methods, and the topic-based methods. Finally, we outline some
other tag-related works and future challenges of tag-aware recommendation
algorithms.Comment: 19 pages, 3 figure
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