13,163 research outputs found
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
Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem
In this paper, we develop a Bayesian evidence maximization framework to solve
the sparse non-negative least squares (S-NNLS) problem. We introduce a family
of probability densities referred to as the Rectified Gaussian Scale Mixture
(R- GSM) to model the sparsity enforcing prior distribution for the solution.
The R-GSM prior encompasses a variety of heavy-tailed densities such as the
rectified Laplacian and rectified Student- t distributions with a proper choice
of the mixing density. We utilize the hierarchical representation induced by
the R-GSM prior and develop an evidence maximization framework based on the
Expectation-Maximization (EM) algorithm. Using the EM based method, we estimate
the hyper-parameters and obtain a point estimate for the solution. We refer to
the proposed method as rectified sparse Bayesian learning (R-SBL). We provide
four R- SBL variants that offer a range of options for computational complexity
and the quality of the E-step computation. These methods include the Markov
chain Monte Carlo EM, linear minimum mean-square-error estimation, approximate
message passing and a diagonal approximation. Using numerical experiments, we
show that the proposed R-SBL method outperforms existing S-NNLS solvers in
terms of both signal and support recovery performance, and is also very robust
against the structure of the design matrix.Comment: Under Review by IEEE Transactions on Signal Processin
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