809 research outputs found
Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry
Convex optimization is a well-established research area with applications in
almost all fields. Over the decades, multiple approaches have been proposed to
solve convex programs. The development of interior-point methods allowed
solving a more general set of convex programs known as semi-definite programs
and second-order cone programs. However, it has been established that these
methods are excessively slow for high dimensions, i.e., they suffer from the
curse of dimensionality. On the other hand, optimization algorithms on manifold
have shown great ability in finding solutions to nonconvex problems in
reasonable time. This paper is interested in solving a subset of convex
optimization using a different approach. The main idea behind Riemannian
optimization is to view the constrained optimization problem as an
unconstrained one over a restricted search space. The paper introduces three
manifolds to solve convex programs under particular box constraints. The
manifolds, called the doubly stochastic, symmetric and the definite multinomial
manifolds, generalize the simplex also known as the multinomial manifold. The
proposed manifolds and algorithms are well-adapted to solving convex programs
in which the variable of interest is a multidimensional probability
distribution function. Theoretical analysis and simulation results testify the
efficiency of the proposed method over state of the art methods. In particular,
they reveal that the proposed framework outperforms conventional generic and
specialized solvers, especially in high dimensions
Efficient Correlated Topic Modeling with Topic Embedding
Correlated topic modeling has been limited to small model and problem sizes
due to their high computational cost and poor scaling. In this paper, we
propose a new model which learns compact topic embeddings and captures topic
correlations through the closeness between the topic vectors. Our method
enables efficient inference in the low-dimensional embedding space, reducing
previous cubic or quadratic time complexity to linear w.r.t the topic size. We
further speedup variational inference with a fast sampler to exploit sparsity
of topic occurrence. Extensive experiments show that our approach is capable of
handling model and data scales which are several orders of magnitude larger
than existing correlation results, without sacrificing modeling quality by
providing competitive or superior performance in document classification and
retrieval.Comment: KDD 2017 oral. The first two authors contributed equall
Probabilistic Clustering Using Maximal Matrix Norm Couplings
In this paper, we present a local information theoretic approach to
explicitly learn probabilistic clustering of a discrete random variable. Our
formulation yields a convex maximization problem for which it is NP-hard to
find the global optimum. In order to algorithmically solve this optimization
problem, we propose two relaxations that are solved via gradient ascent and
alternating maximization. Experiments on the MSR Sentence Completion Challenge,
MovieLens 100K, and Reuters21578 datasets demonstrate that our approach is
competitive with existing techniques and worthy of further investigation.Comment: Presented at 56th Annual Allerton Conference on Communication,
Control, and Computing, 201
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