1,616 research outputs found
Approximation Algorithms for Bregman Co-clustering and Tensor Clustering
In the past few years powerful generalizations to the Euclidean k-means
problem have been made, such as Bregman clustering [7], co-clustering (i.e.,
simultaneous clustering of rows and columns of an input matrix) [9,18], and
tensor clustering [8,34]. Like k-means, these more general problems also suffer
from the NP-hardness of the associated optimization. Researchers have developed
approximation algorithms of varying degrees of sophistication for k-means,
k-medians, and more recently also for Bregman clustering [2]. However, there
seem to be no approximation algorithms for Bregman co- and tensor clustering.
In this paper we derive the first (to our knowledge) guaranteed methods for
these increasingly important clustering settings. Going beyond Bregman
divergences, we also prove an approximation factor for tensor clustering with
arbitrary separable metrics. Through extensive experiments we evaluate the
characteristics of our method, and show that it also has practical impact.Comment: 18 pages; improved metric cas
Regression on fixed-rank positive semidefinite matrices: a Riemannian approach
The paper addresses the problem of learning a regression model parameterized
by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear
nature of the search space and on scalability to high-dimensional problems. The
mathematical developments rely on the theory of gradient descent algorithms
adapted to the Riemannian geometry that underlies the set of fixed-rank
positive semidefinite matrices. In contrast with previous contributions in the
literature, no restrictions are imposed on the range space of the learned
matrix. The resulting algorithms maintain a linear complexity in the problem
size and enjoy important invariance properties. We apply the proposed
algorithms to the problem of learning a distance function parameterized by a
positive semidefinite matrix. Good performance is observed on classical
benchmarks
-MLE: A fast algorithm for learning statistical mixture models
We describe -MLE, a fast and efficient local search algorithm for learning
finite statistical mixtures of exponential families such as Gaussian mixture
models. Mixture models are traditionally learned using the
expectation-maximization (EM) soft clustering technique that monotonically
increases the incomplete (expected complete) likelihood. Given prescribed
mixture weights, the hard clustering -MLE algorithm iteratively assigns data
to the most likely weighted component and update the component models using
Maximum Likelihood Estimators (MLEs). Using the duality between exponential
families and Bregman divergences, we prove that the local convergence of the
complete likelihood of -MLE follows directly from the convergence of a dual
additively weighted Bregman hard clustering. The inner loop of -MLE can be
implemented using any -means heuristic like the celebrated Lloyd's batched
or Hartigan's greedy swap updates. We then show how to update the mixture
weights by minimizing a cross-entropy criterion that implies to update weights
by taking the relative proportion of cluster points, and reiterate the mixture
parameter update and mixture weight update processes until convergence. Hard EM
is interpreted as a special case of -MLE when both the component update and
the weight update are performed successively in the inner loop. To initialize
-MLE, we propose -MLE++, a careful initialization of -MLE guaranteeing
probabilistically a global bound on the best possible complete likelihood.Comment: 31 pages, Extend preliminary paper presented at IEEE ICASSP 201
The Burbea-Rao and Bhattacharyya centroids
We study the centroid with respect to the class of information-theoretic
Burbea-Rao divergences that generalize the celebrated Jensen-Shannon divergence
by measuring the non-negative Jensen difference induced by a strictly convex
and differentiable function. Although those Burbea-Rao divergences are
symmetric by construction, they are not metric since they fail to satisfy the
triangle inequality. We first explain how a particular symmetrization of
Bregman divergences called Jensen-Bregman distances yields exactly those
Burbea-Rao divergences. We then proceed by defining skew Burbea-Rao
divergences, and show that skew Burbea-Rao divergences amount in limit cases to
compute Bregman divergences. We then prove that Burbea-Rao centroids are
unique, and can be arbitrarily finely approximated by a generic iterative
concave-convex optimization algorithm with guaranteed convergence property. In
the second part of the paper, we consider the Bhattacharyya distance that is
commonly used to measure overlapping degree of probability distributions. We
show that Bhattacharyya distances on members of the same statistical
exponential family amount to calculate a Burbea-Rao divergence in disguise.
Thus we get an efficient algorithm for computing the Bhattacharyya centroid of
a set of parametric distributions belonging to the same exponential families,
improving over former specialized methods found in the literature that were
limited to univariate or "diagonal" multivariate Gaussians. To illustrate the
performance of our Bhattacharyya/Burbea-Rao centroid algorithm, we present
experimental performance results for -means and hierarchical clustering
methods of Gaussian mixture models.Comment: 13 page
- …