21,206 research outputs found
Unfolding Latent Tree Structures using 4th Order Tensors
Discovering the latent structure from many observed variables is an important
yet challenging learning task. Existing approaches for discovering latent
structures often require the unknown number of hidden states as an input. In
this paper, we propose a quartet based approach which is \emph{agnostic} to
this number. The key contribution is a novel rank characterization of the
tensor associated with the marginal distribution of a quartet. This
characterization allows us to design a \emph{nuclear norm} based test for
resolving quartet relations. We then use the quartet test as a subroutine in a
divide-and-conquer algorithm for recovering the latent tree structure. Under
mild conditions, the algorithm is consistent and its error probability decays
exponentially with increasing sample size. We demonstrate that the proposed
approach compares favorably to alternatives. In a real world stock dataset, it
also discovers meaningful groupings of variables, and produces a model that
fits the data better
Kernel Bayes' rule
A nonparametric kernel-based method for realizing Bayes' rule is proposed,
based on representations of probabilities in reproducing kernel Hilbert spaces.
Probabilities are uniquely characterized by the mean of the canonical map to
the RKHS. The prior and conditional probabilities are expressed in terms of
RKHS functions of an empirical sample: no explicit parametric model is needed
for these quantities. The posterior is likewise an RKHS mean of a weighted
sample. The estimator for the expectation of a function of the posterior is
derived, and rates of consistency are shown. Some representative applications
of the kernel Bayes' rule are presented, including Baysian computation without
likelihood and filtering with a nonparametric state-space model.Comment: 27 pages, 5 figure
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