1,837 research outputs found
Sequential importance sampling for multiway tables
We describe an algorithm for the sequential sampling of entries in multiway
contingency tables with given constraints. The algorithm can be used for
computations in exact conditional inference. To justify the algorithm, a theory
relates sampling values at each step to properties of the associated toric
ideal using computational commutative algebra. In particular, the property of
interval cell counts at each step is related to exponents on lead
indeterminates of a lexicographic Gr\"{o}bner basis. Also, the approximation of
integer programming by linear programming for sampling is related to initial
terms of a toric ideal. We apply the algorithm to examples of contingency
tables which appear in the social and medical sciences. The numerical results
demonstrate that the theory is applicable and that the algorithm performs well.Comment: Published at http://dx.doi.org/10.1214/009053605000000822 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Tensor-on-tensor regression
We propose a framework for the linear prediction of a multi-way array (i.e.,
a tensor) from another multi-way array of arbitrary dimension, using the
contracted tensor product. This framework generalizes several existing
approaches, including methods to predict a scalar outcome from a tensor, a
matrix from a matrix, or a tensor from a scalar. We describe an approach that
exploits the multiway structure of both the predictors and the outcomes by
restricting the coefficients to have reduced CP-rank. We propose a general and
efficient algorithm for penalized least-squares estimation, which allows for a
ridge (L_2) penalty on the coefficients. The objective is shown to give the
mode of a Bayesian posterior, which motivates a Gibbs sampling algorithm for
inference. We illustrate the approach with an application to facial image data.
An R package is available at https://github.com/lockEF/MultiwayRegression .Comment: 33 pages, 3 figure
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
Dictionary-based Tensor Canonical Polyadic Decomposition
To ensure interpretability of extracted sources in tensor decomposition, we
introduce in this paper a dictionary-based tensor canonical polyadic
decomposition which enforces one factor to belong exactly to a known
dictionary. A new formulation of sparse coding is proposed which enables high
dimensional tensors dictionary-based canonical polyadic decomposition. The
benefits of using a dictionary in tensor decomposition models are explored both
in terms of parameter identifiability and estimation accuracy. Performances of
the proposed algorithms are evaluated on the decomposition of simulated data
and the unmixing of hyperspectral images
Tensor decomposition with generalized lasso penalties
We present an approach for penalized tensor decomposition (PTD) that
estimates smoothly varying latent factors in multi-way data. This generalizes
existing work on sparse tensor decomposition and penalized matrix
decompositions, in a manner parallel to the generalized lasso for regression
and smoothing problems. Our approach presents many nontrivial challenges at the
intersection of modeling and computation, which are studied in detail. An
efficient coordinate-wise optimization algorithm for (PTD) is presented, and
its convergence properties are characterized. The method is applied both to
simulated data and real data on flu hospitalizations in Texas. These results
show that our penalized tensor decomposition can offer major improvements on
existing methods for analyzing multi-way data that exhibit smooth spatial or
temporal features
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