591 research outputs found
Characterizations of Decomposable Dependency Models
Decomposable dependency models possess a number of interesting and useful
properties. This paper presents new characterizations of decomposable models in
terms of independence relationships, which are obtained by adding a single
axiom to the well-known set characterizing dependency models that are
isomorphic to undirected graphs. We also briefly discuss a potential
application of our results to the problem of learning graphical models from
data.Comment: See http://www.jair.org/ for any accompanying file
Characterizations of Decomposable Dependency Models
Decomposable dependency models possess a number of interesting and useful proper-
ties. This paper presents new characterizations of decomposable models in terms of in-
dependence relationships, which are obtained by adding a single axiom to the well-known
set characterizing dependency models that are isomorphic to undirected graphs. We also
brie
y discuss a potential application of our results to the problem of learning graphical
models from data.Spanish Comisión Interministerial de Ciencia y Tec-
nologÃa (CICYT) TIC96-078
Estimation with Norm Regularization
Analysis of non-asymptotic estimation error and structured statistical
recovery based on norm regularized regression, such as Lasso, needs to consider
four aspects: the norm, the loss function, the design matrix, and the noise
model. This paper presents generalizations of such estimation error analysis on
all four aspects compared to the existing literature. We characterize the
restricted error set where the estimation error vector lies, establish
relations between error sets for the constrained and regularized problems, and
present an estimation error bound applicable to any norm. Precise
characterizations of the bound is presented for isotropic as well as
anisotropic subGaussian design matrices, subGaussian noise models, and convex
loss functions, including least squares and generalized linear models. Generic
chaining and associated results play an important role in the analysis. A key
result from the analysis is that the sample complexity of all such estimators
depends on the Gaussian width of a spherical cap corresponding to the
restricted error set. Further, once the number of samples crosses the
required sample complexity, the estimation error decreases as
, where depends on the Gaussian width of the unit norm
ball.Comment: Fixed technical issues. Generalized some result
Tensor decompositions for learning latent variable models
This work considers a computationally and statistically efficient parameter
estimation method for a wide class of latent variable models---including
Gaussian mixture models, hidden Markov models, and latent Dirichlet
allocation---which exploits a certain tensor structure in their low-order
observable moments (typically, of second- and third-order). Specifically,
parameter estimation is reduced to the problem of extracting a certain
(orthogonal) decomposition of a symmetric tensor derived from the moments; this
decomposition can be viewed as a natural generalization of the singular value
decomposition for matrices. Although tensor decompositions are generally
intractable to compute, the decomposition of these specially structured tensors
can be efficiently obtained by a variety of approaches, including power
iterations and maximization approaches (similar to the case of matrices). A
detailed analysis of a robust tensor power method is provided, establishing an
analogue of Wedin's perturbation theorem for the singular vectors of matrices.
This implies a robust and computationally tractable estimation approach for
several popular latent variable models
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