513 research outputs found
Robustly Learning Mixtures of Arbitrary Gaussians
We give a polynomial-time algorithm for the problem of robustly estimating a
mixture of arbitrary Gaussians in , for any fixed , in the
presence of a constant fraction of arbitrary corruptions. This resolves the
main open problem in several previous works on algorithmic robust statistics,
which addressed the special cases of robustly estimating (a) a single Gaussian,
(b) a mixture of TV-distance separated Gaussians, and (c) a uniform mixture of
two Gaussians. Our main tools are an efficient \emph{partial clustering}
algorithm that relies on the sum-of-squares method, and a novel \emph{tensor
decomposition} algorithm that allows errors in both Frobenius norm and low-rank
terms.Comment: This version extends the previous one to yield 1) robust proper
learning algorithm with poly(eps) error and 2) an information theoretic
argument proving that the same algorithms in fact also yield parameter
recovery guarantees. The updates are included in Sections 7,8, and 9 and the
main result from the previous version (Thm 1.4) is presented and proved in
Section
Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity
A general framework for solving image inverse problems is introduced in this
paper. The approach is based on Gaussian mixture models, estimated via a
computationally efficient MAP-EM algorithm. A dual mathematical interpretation
of the proposed framework with structured sparse estimation is described, which
shows that the resulting piecewise linear estimate stabilizes the estimation
when compared to traditional sparse inverse problem techniques. This
interpretation also suggests an effective dictionary motivated initialization
for the MAP-EM algorithm. We demonstrate that in a number of image inverse
problems, including inpainting, zooming, and deblurring, the same algorithm
produces either equal, often significantly better, or very small margin worse
results than the best published ones, at a lower computational cost.Comment: 30 page
Convex Clustering via Optimal Mass Transport
We consider approximating distributions within the framework of optimal mass
transport and specialize to the problem of clustering data sets. Distances
between distributions are measured in the Wasserstein metric. The main problem
we consider is that of approximating sample distributions by ones with sparse
support. This provides a new viewpoint to clustering. We propose different
relaxations of a cardinality function which penalizes the size of the support
set. We establish that a certain relaxation provides the tightest convex lower
approximation to the cardinality penalty. We compare the performance of
alternative relaxations on a numerical study on clustering.Comment: 12 pages, 12 figure
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