4,202 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
Periodic and homoclinic solutions of the modified 2+1 Chiral model
We use algebraic Backlund transformations (BTs) to construct explicit
solutions of the modified 2+1 chiral model from to SU(n), where
is a 2-torus. Algebraic BTs are parameterized by (poles) and
holomorphic maps from to Gr. We apply B\"acklund
transformations with carefully chosen poles and 's to construct infinitely
many solutions of the 2+1 chiral model that are (i) doubly periodic in space
variables and periodic in time, i.e., triply periodic, (ii) homoclinic in the
sense that the solution has the same stationary limit as and is tangent to a stable linear mode of as and
is tangent to an unstable mode of as .Comment: 17 page
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