4,953 research outputs found
Singular random matrix decompositions: distributions.
Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and Polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and Pseudo-Wishart generalized singular and non-singular distributions. We present a particular example for the Karhunen-Lòeve decomposition. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution
Multivariate Analysis of Mixed Data: The R Package PCAmixdata
Mixed data arise when observations are described by a mixture of numerical
and categorical variables. The R package PCAmixdata extends standard
multivariate analysis methods to incorporate this type of data. The key
techniques/methods included in the package are principal component analysis for
mixed data (PCAmix), varimax-like orthogonal rotation for PCAmix, and multiple
factor analysis for mixed multi-table data. This paper gives a synthetic
presentation of the three algorithms with details to help the user understand
graphical and numerical outputs of the corresponding R functions. The three
main methods are illustrated on a real dataset composed of four data tables
characterizing living conditions in different municipalities in the Gironde
region of southwest France
SINGULAR RANDOM MATRIX DECOMPOSITIONS: DISTRIBUTIONS.
Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and Polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and Pseudo-Wishart generalized singular and non-singular distributions. We present a particular example for the Karhunen-Lòeve decomposition. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution.
Efficient Orthogonal Tensor Decomposition, with an Application to Latent Variable Model Learning
Decomposing tensors into orthogonal factors is a well-known task in
statistics, machine learning, and signal processing. We study orthogonal outer
product decompositions where the factors in the summands in the decomposition
are required to be orthogonal across summands, by relating this orthogonal
decomposition to the singular value decompositions of the flattenings. We show
that it is a non-trivial assumption for a tensor to have such an orthogonal
decomposition, and we show that it is unique (up to natural symmetries) in case
it exists, in which case we also demonstrate how it can be efficiently and
reliably obtained by a sequence of singular value decompositions. We
demonstrate how the factoring algorithm can be applied for parameter
identification in latent variable and mixture models
Nonparametric Estimation of Multi-View Latent Variable Models
Spectral methods have greatly advanced the estimation of latent variable
models, generating a sequence of novel and efficient algorithms with strong
theoretical guarantees. However, current spectral algorithms are largely
restricted to mixtures of discrete or Gaussian distributions. In this paper, we
propose a kernel method for learning multi-view latent variable models,
allowing each mixture component to be nonparametric. The key idea of the method
is to embed the joint distribution of a multi-view latent variable into a
reproducing kernel Hilbert space, and then the latent parameters are recovered
using a robust tensor power method. We establish that the sample complexity for
the proposed method is quadratic in the number of latent components and is a
low order polynomial in the other relevant parameters. Thus, our non-parametric
tensor approach to learning latent variable models enjoys good sample and
computational efficiencies. Moreover, the non-parametric tensor power method
compares favorably to EM algorithm and other existing spectral algorithms in
our experiments
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