12,593 research outputs found
A new ADMM algorithm for the Euclidean median and its application to robust patch regression
The Euclidean Median (EM) of a set of points in an Euclidean space
is the point x minimizing the (weighted) sum of the Euclidean distances of x to
the points in . While there exits no closed-form expression for the EM,
it can nevertheless be computed using iterative methods such as the Wieszfeld
algorithm. The EM has classically been used as a robust estimator of centrality
for multivariate data. It was recently demonstrated that the EM can be used to
perform robust patch-based denoising of images by generalizing the popular
Non-Local Means algorithm. In this paper, we propose a novel algorithm for
computing the EM (and its box-constrained counterpart) using variable splitting
and the method of augmented Lagrangian. The attractive feature of this approach
is that the subproblems involved in the ADMM-based optimization of the
augmented Lagrangian can be resolved using simple closed-form projections. The
proposed ADMM solver is used for robust patch-based image denoising and is
shown to exhibit faster convergence compared to an existing solver.Comment: 5 pages, 3 figures, 1 table. To appear in Proc. IEEE International
Conference on Acoustics, Speech, and Signal Processing, April 19-24, 201
Bayesian nonparametric multivariate convex regression
In many applications, such as economics, operations research and
reinforcement learning, one often needs to estimate a multivariate regression
function f subject to a convexity constraint. For example, in sequential
decision processes the value of a state under optimal subsequent decisions may
be known to be convex or concave. We propose a new Bayesian nonparametric
multivariate approach based on characterizing the unknown regression function
as the max of a random collection of unknown hyperplanes. This specification
induces a prior with large support in a Kullback-Leibler sense on the space of
convex functions, while also leading to strong posterior consistency. Although
we assume that f is defined over R^p, we show that this model has a convergence
rate of log(n)^{-1} n^{-1/(d+2)} under the empirical L2 norm when f actually
maps a d dimensional linear subspace to R. We design an efficient reversible
jump MCMC algorithm for posterior computation and demonstrate the methods
through application to value function approximation
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