1,962 research outputs found
Regularized Maximum Likelihood Estimation and Feature Selection in Mixtures-of-Experts Models
Mixture of Experts (MoE) are successful models for modeling heterogeneous
data in many statistical learning problems including regression, clustering and
classification. Generally fitted by maximum likelihood estimation via the
well-known EM algorithm, their application to high-dimensional problems is
still therefore challenging. We consider the problem of fitting and feature
selection in MoE models, and propose a regularized maximum likelihood
estimation approach that encourages sparse solutions for heterogeneous
regression data models with potentially high-dimensional predictors. Unlike
state-of-the art regularized MLE for MoE, the proposed modelings do not require
an approximate of the penalty function. We develop two hybrid EM algorithms: an
Expectation-Majorization-Maximization (EM/MM) algorithm, and an EM algorithm
with coordinate ascent algorithm. The proposed algorithms allow to
automatically obtaining sparse solutions without thresholding, and avoid matrix
inversion by allowing univariate parameter updates. An experimental study shows
the good performance of the algorithms in terms of recovering the actual sparse
solutions, parameter estimation, and clustering of heterogeneous regression
data
Regularized Optimal Transport and the Rot Mover's Distance
This paper presents a unified framework for smooth convex regularization of
discrete optimal transport problems. In this context, the regularized optimal
transport turns out to be equivalent to a matrix nearness problem with respect
to Bregman divergences. Our framework thus naturally generalizes a previously
proposed regularization based on the Boltzmann-Shannon entropy related to the
Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We
call the regularized optimal transport distance the rot mover's distance in
reference to the classical earth mover's distance. We develop two generic
schemes that we respectively call the alternate scaling algorithm and the
non-negative alternate scaling algorithm, to compute efficiently the
regularized optimal plans depending on whether the domain of the regularizer
lies within the non-negative orthant or not. These schemes are based on
Dykstra's algorithm with alternate Bregman projections, and further exploit the
Newton-Raphson method when applied to separable divergences. We enhance the
separable case with a sparse extension to deal with high data dimensions. We
also instantiate our proposed framework and discuss the inherent specificities
for well-known regularizers and statistical divergences in the machine learning
and information geometry communities. Finally, we demonstrate the merits of our
methods with experiments using synthetic data to illustrate the effect of
different regularizers and penalties on the solutions, as well as real-world
data for a pattern recognition application to audio scene classification
One-step estimator paths for concave regularization
The statistics literature of the past 15 years has established many favorable
properties for sparse diminishing-bias regularization: techniques which can
roughly be understood as providing estimation under penalty functions spanning
the range of concavity between and norms. However, lasso
-regularized estimation remains the standard tool for industrial `Big
Data' applications because of its minimal computational cost and the presence
of easy-to-apply rules for penalty selection. In response, this article
proposes a simple new algorithm framework that requires no more computation
than a lasso path: the path of one-step estimators (POSE) does penalized
regression estimation on a grid of decreasing penalties, but adapts
coefficient-specific weights to decrease as a function of the coefficient
estimated in the previous path step. This provides sparse diminishing-bias
regularization at no extra cost over the fastest lasso algorithms. Moreover,
our `gamma lasso' implementation of POSE is accompanied by a reliable heuristic
for the fit degrees of freedom, so that standard information criteria can be
applied in penalty selection. We also provide novel results on the distance
between weighted- and penalized predictors; this allows us to build
intuition about POSE and other diminishing-bias regularization schemes. The
methods and results are illustrated in extensive simulations and in application
of logistic regression to evaluating the performance of hockey players.Comment: Data and code are in the gamlr package for R. Supplemental appendix
is at https://github.com/TaddyLab/pose/raw/master/paper/supplemental.pd
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