669 research outputs found
Online Tensor Methods for Learning Latent Variable Models
We introduce an online tensor decomposition based approach for two latent
variable modeling problems namely, (1) community detection, in which we learn
the latent communities that the social actors in social networks belong to, and
(2) topic modeling, in which we infer hidden topics of text articles. We
consider decomposition of moment tensors using stochastic gradient descent. We
conduct optimization of multilinear operations in SGD and avoid directly
forming the tensors, to save computational and storage costs. We present
optimized algorithm in two platforms. Our GPU-based implementation exploits the
parallelism of SIMD architectures to allow for maximum speed-up by a careful
optimization of storage and data transfer, whereas our CPU-based implementation
uses efficient sparse matrix computations and is suitable for large sparse
datasets. For the community detection problem, we demonstrate accuracy and
computational efficiency on Facebook, Yelp and DBLP datasets, and for the topic
modeling problem, we also demonstrate good performance on the New York Times
dataset. We compare our results to the state-of-the-art algorithms such as the
variational method, and report a gain of accuracy and a gain of several orders
of magnitude in the execution time.Comment: JMLR 201
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
Jacobi iterations for canonical dependence analysis
International audienceIn this manuscript we will study the advantages of Jacobi iterations to solve the problem of Canonical Dependence Analysis. Canonical Dependence Analysis can be seen as an extension of the Canonical Correlation Analysis where correlation measures are replaced by measures of higher order statistical dependencies. We will show the benefits of choosing an algorithm that exploits the manifold structure on which the optimisation problem can be formulated and contrast our results with the joint blind source separation algorithm that optimises the criterion in its ambient space. A major advantage of the proposed algorithm is the capability of identifying a linear mixture when multiple observation sets are available containing variables that are linearly dependent between the sets, independent within the sets and contaminated with non-Gaussian independent noise. Performance analysis reveals at least linear convergence speed as a function of the number of sweeps
Lecture Notes of Tensor Network Contractions
Tensor network (TN), a young mathematical tool of high vitality and great
potential, has been undergoing extremely rapid developments in the last two
decades, gaining tremendous success in condensed matter physics, atomic
physics, quantum information science, statistical physics, and so on. In this
lecture notes, we focus on the contraction algorithms of TN as well as some of
the applications to the simulations of quantum many-body systems. Starting from
basic concepts and definitions, we first explain the relations between TN and
physical problems, including the TN representations of classical partition
functions, quantum many-body states (by matrix product state, tree TN, and
projected entangled pair state), time evolution simulations, etc. These
problems, which are challenging to solve, can be transformed to TN contraction
problems. We present then several paradigm algorithms based on the ideas of the
numerical renormalization group and/or boundary states, including density
matrix renormalization group, time-evolving block decimation,
coarse-graining/corner tensor renormalization group, and several distinguished
variational algorithms. Finally, we revisit the TN approaches from the
perspective of multi-linear algebra (also known as tensor algebra or tensor
decompositions) and quantum simulation. Despite the apparent differences in the
ideas and strategies of different TN algorithms, we aim at revealing the
underlying relations and resemblances in order to present a systematic picture
to understand the TN contraction approaches.Comment: 134 pages, 68 figures. In this version, the manuscript has been
changed into the format of book; new sections about tensor network and
quantum circuits have been adde
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Factor Analysis of Data Matrices: New Theoretical and Computational Aspects With Applications
The classical fitting problem in exploratory factor analysis (EFA) is to find estimates for the factor loadings matrix and the matrix of unique factor variances which give the best fit to the sample covariance or correlation matrix with respect to some goodness-of-fit criterion. Predicted factor scores can be obtained as a function of these estimates and the data. In this thesis, the EFA model is considered as a specific data matrix decomposition with fixed unknown matrix parameters. Fitting the EFA model directly to the data yields simultaneous solutions for both loadings and factor scores. Several new algorithms are introduced for the least squares and weighted least squares estimation of all EFA model unknowns. The numerical procedures are based on the singular value decomposition, facilitate the estimation of both common and unique factor scores, and work equally well when the number of variables exceeds the number of available observations.
Like EFA, noisy independent component analysis (ICA) is a technique for reduction of the data dimensionality in which the interrelationships among the observed variables are explained in terms of a much smaller number of latent factors. The key difference between EFA and noisy ICA is that in the latter model the common factors are assumed to be both independent and non-normal. In contrast to EFA, there is no rotational indeterminacy in noisy ICA. In this thesis, noisy ICA is viewed as a method of factor rotation in EFA. Starting from an initial EFA solution, an orthogonal rotation matrix is sought that minimizes the dependence between the common factors. The idea of rotating the scores towards independence is also employed in three-mode factor analysis to analyze data sets having a three-way structure.
The new theoretical and computational aspects contained in this thesis are illustrated by means of several examples with real and artificial data
Computational Studies of Quantum Spin Systems
These lecture notes introduce quantum spin systems and several computational
methods for studying their ground-state and finite-temperature properties.
Symmetry-breaking and critical phenomena are first discussed in the simpler
setting of Monte Carlo studies of classical spin systems, to illustrate
finite-size scaling at continuous and first-order phase transitions. Exact
diagonalization and quantum Monte Carlo (stochastic series expansion)
algorithms and their computer implementations are then discussed in detail.
Applications of the methods are illustrated by results for some of the most
essential models in quantum magnetism, such as the S=1/2 Heisenberg
antiferromagnet in one and two dimensions, as well as extended models useful
for studying quantum phase transitions between antiferromagnetic and
magnetically disordered states.Comment: 207 pages, 91 figures. Lecture notes for course given at the 14th
Training Course in Physics of Strongly Correlated Systems, Salerno (Vietri
sul Mare), Italy, in October 200
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