343 research outputs found
Sliced Wasserstein Distance for Learning Gaussian Mixture Models
Gaussian mixture models (GMM) are powerful parametric tools with many
applications in machine learning and computer vision. Expectation maximization
(EM) is the most popular algorithm for estimating the GMM parameters. However,
EM guarantees only convergence to a stationary point of the log-likelihood
function, which could be arbitrarily worse than the optimal solution. Inspired
by the relationship between the negative log-likelihood function and the
Kullback-Leibler (KL) divergence, we propose an alternative formulation for
estimating the GMM parameters using the sliced Wasserstein distance, which
gives rise to a new algorithm. Specifically, we propose minimizing the
sliced-Wasserstein distance between the mixture model and the data distribution
with respect to the GMM parameters. In contrast to the KL-divergence, the
energy landscape for the sliced-Wasserstein distance is more well-behaved and
therefore more suitable for a stochastic gradient descent scheme to obtain the
optimal GMM parameters. We show that our formulation results in parameter
estimates that are more robust to random initializations and demonstrate that
it can estimate high-dimensional data distributions more faithfully than the EM
algorithm
Max-Sliced Wasserstein Distance and its use for GANs
Generative adversarial nets (GANs) and variational auto-encoders have
significantly improved our distribution modeling capabilities, showing promise
for dataset augmentation, image-to-image translation and feature learning.
However, to model high-dimensional distributions, sequential training and
stacked architectures are common, increasing the number of tunable
hyper-parameters as well as the training time. Nonetheless, the sample
complexity of the distance metrics remains one of the factors affecting GAN
training. We first show that the recently proposed sliced Wasserstein distance
has compelling sample complexity properties when compared to the Wasserstein
distance. To further improve the sliced Wasserstein distance we then analyze
its `projection complexity' and develop the max-sliced Wasserstein distance
which enjoys compelling sample complexity while reducing projection complexity,
albeit necessitating a max estimation. We finally illustrate that the proposed
distance trains GANs on high-dimensional images up to a resolution of 256x256
easily.Comment: Accepted to CVPR 201
Solving general elliptical mixture models through an approximate Wasserstein manifold
We address the estimation problem for general finite mixture models, with a
particular focus on the elliptical mixture models (EMMs). Compared to the
widely adopted Kullback-Leibler divergence, we show that the Wasserstein
distance provides a more desirable optimisation space. We thus provide a stable
solution to the EMMs that is both robust to initialisations and reaches a
superior optimum by adaptively optimising along a manifold of an approximate
Wasserstein distance. To this end, we first provide a unifying account of
computable and identifiable EMMs, which serves as a basis to rigorously address
the underpinning optimisation problem. Due to a probability constraint, solving
this problem is extremely cumbersome and unstable, especially under the
Wasserstein distance. To relieve this issue, we introduce an efficient
optimisation method on a statistical manifold defined under an approximate
Wasserstein distance, which allows for explicit metrics and computable
operations, thus significantly stabilising and improving the EMM estimation. We
further propose an adaptive method to accelerate the convergence. Experimental
results demonstrate the excellent performance of the proposed EMM solver.Comment: This work has been accepted to AAAI2020. Note that this version also
corrects a small error on the Equation (16) in proo
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