279 research outputs found
Nonnegative approximations of nonnegative tensors
We study the decomposition of a nonnegative tensor into a minimal sum of
outer product of nonnegative vectors and the associated parsimonious naive
Bayes probabilistic model. We show that the corresponding approximation
problem, which is central to nonnegative PARAFAC, will always have optimal
solutions. The result holds for any choice of norms and, under a mild
assumption, even Bregman divergences.Comment: 14 page
The Bregman Variational Dual-Tree Framework
Graph-based methods provide a powerful tool set for many non-parametric
frameworks in Machine Learning. In general, the memory and computational
complexity of these methods is quadratic in the number of examples in the data
which makes them quickly infeasible for moderate to large scale datasets. A
significant effort to find more efficient solutions to the problem has been
made in the literature. One of the state-of-the-art methods that has been
recently introduced is the Variational Dual-Tree (VDT) framework. Despite some
of its unique features, VDT is currently restricted only to Euclidean spaces
where the Euclidean distance quantifies the similarity. In this paper, we
extend the VDT framework beyond the Euclidean distance to more general Bregman
divergences that include the Euclidean distance as a special case. By
exploiting the properties of the general Bregman divergence, we show how the
new framework can maintain all the pivotal features of the VDT framework and
yet significantly improve its performance in non-Euclidean domains. We apply
the proposed framework to different text categorization problems and
demonstrate its benefits over the original VDT.Comment: Appears in Proceedings of the Twenty-Ninth Conference on Uncertainty
in Artificial Intelligence (UAI2013
Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative Matrix Factorization
We propose a class of multiplicative algorithms for Nonnegative Matrix Factorization (NMF) which are robust with respect to noise and outliers. To achieve this, we formulate a new family generalized divergences referred to as the Alpha-Beta-divergences (AB-divergences), which are parameterized by the two tuning parameters, alpha and beta, and smoothly connect the fundamental Alpha-, Beta- and Gamma-divergences. By adjusting these tuning parameters, we show that a wide range of standard and new divergences can be obtained. The corresponding learning algorithms for NMF are shown to integrate and generalize many existing ones, including the Lee-Seung, ISRA (Image Space Reconstruction Algorithm), EMML (Expectation Maximization Maximum Likelihood), Alpha-NMF, and Beta-NMF. Owing to more degrees of freedom in tuning the parameters, the proposed family of AB-multiplicative NMF algorithms is shown to improve robustness with respect to noise and outliers. The analysis illuminates the links of between AB-divergence and other divergences, especially Gamma- and Itakura-Saito divergences
The Diagonalized Newton Algorithm for Nonnegative Matrix Factorization
Non-negative matrix factorization (NMF) has become a popular machine learning
approach to many problems in text mining, speech and image processing,
bio-informatics and seismic data analysis to name a few. In NMF, a matrix of
non-negative data is approximated by the low-rank product of two matrices with
non-negative entries. In this paper, the approximation quality is measured by
the Kullback-Leibler divergence between the data and its low-rank
reconstruction. The existence of the simple multiplicative update (MU)
algorithm for computing the matrix factors has contributed to the success of
NMF. Despite the availability of algorithms showing faster convergence, MU
remains popular due to its simplicity. In this paper, a diagonalized Newton
algorithm (DNA) is proposed showing faster convergence while the implementation
remains simple and suitable for high-rank problems. The DNA algorithm is
applied to various publicly available data sets, showing a substantial speed-up
on modern hardware.Comment: 8 pages + references; International Conference on Learning
Representations, 201
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
Algorithms for nonnegative matrix factorization with the beta-divergence
This paper describes algorithms for nonnegative matrix factorization (NMF)
with the beta-divergence (beta-NMF). The beta-divergence is a family of cost
functions parametrized by a single shape parameter beta that takes the
Euclidean distance, the Kullback-Leibler divergence and the Itakura-Saito
divergence as special cases (beta = 2,1,0, respectively). The proposed
algorithms are based on a surrogate auxiliary function (a local majorization of
the criterion function). We first describe a majorization-minimization (MM)
algorithm that leads to multiplicative updates, which differ from standard
heuristic multiplicative updates by a beta-dependent power exponent. The
monotonicity of the heuristic algorithm can however be proven for beta in (0,1)
using the proposed auxiliary function. Then we introduce the concept of
majorization-equalization (ME) algorithm which produces updates that move along
constant level sets of the auxiliary function and lead to larger steps than MM.
Simulations on synthetic and real data illustrate the faster convergence of the
ME approach. The paper also describes how the proposed algorithms can be
adapted to two common variants of NMF : penalized NMF (i.e., when a penalty
function of the factors is added to the criterion function) and convex-NMF
(when the dictionary is assumed to belong to a known subspace).Comment: \`a para\^itre dans Neural Computatio
Topological Data Analysis with Bregman Divergences
Given a finite set in a metric space, the topological analysis generalizes
hierarchical clustering using a 1-parameter family of homology groups to
quantify connectivity in all dimensions. The connectivity is compactly
described by the persistence diagram. One limitation of the current framework
is the reliance on metric distances, whereas in many practical applications
objects are compared by non-metric dissimilarity measures. Examples are the
Kullback-Leibler divergence, which is commonly used for comparing text and
images, and the Itakura-Saito divergence, popular for speech and sound. These
are two members of the broad family of dissimilarities called Bregman
divergences.
We show that the framework of topological data analysis can be extended to
general Bregman divergences, widening the scope of possible applications. In
particular, we prove that appropriately generalized Cech and Delaunay (alpha)
complexes capture the correct homotopy type, namely that of the corresponding
union of Bregman balls. Consequently, their filtrations give the correct
persistence diagram, namely the one generated by the uniformly growing Bregman
balls. Moreover, we show that unlike the metric setting, the filtration of
Vietoris-Rips complexes may fail to approximate the persistence diagram. We
propose algorithms to compute the thus generalized Cech, Vietoris-Rips and
Delaunay complexes and experimentally test their efficiency. Lastly, we explain
their surprisingly good performance by making a connection with discrete Morse
theory
- …