3,974 research outputs found
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
Harmonic mappings valued in the Wasserstein space
We propose a definition of the Dirichlet energy (which is roughly speaking
the integral of the square of the gradient) for mappings mu : Omega -> (P(D),
W\_2) defined over a subset Omega of R^p and valued in the space P(D) of
probability measures on a compact convex subset D of R^q endowed with the
quadratic Wasserstein distance. Our definition relies on a straightforward
generalization of the Benamou-Brenier formula (already introduced by Brenier)
but is also equivalent to the definition of Koorevaar, Schoen and Jost as limit
of approximate Dirichlet energies, and to the definition of Reshetnyak of
Sobolev spaces valued in metric spaces. We study harmonic mappings, i.e.
minimizers of the Dirichlet energy provided that the values on the boundary d
Omega are fixed. The notion of constant-speed geodesics in the Wasserstein
space is recovered by taking for Omega a segment of R. As the Wasserstein space
(P(D), W\_2) is positively curved in the sense of Alexandrov we cannot apply
the theory of Koorevaar, Schoen and Jost and we use instead arguments based on
optimal transport. We manage to get existence of harmonic mappings provided
that the boundary values are Lipschitz on d Omega, uniqueness is an open
question. If Omega is a segment of R, it is known that a curve valued in the
Wasserstein space P(D) can be seen as a superposition of curves valued in D. We
show that it is no longer the case in higher dimensions: a generic mapping
Omega -> P(D) cannot be represented as the superposition of mappings Omega ->
D. We are able to show the validity of a maximum principle: the composition
F(mu) of a function F : P(D) -> R convex along generalized geodesics and a
harmonic mapping mu : Omega -> P(D) is a subharmonic real-valued function. We
also study the special case where we restrict ourselves to a given family of
elliptically contoured distributions (a finite-dimensional and geodesically
convex submanifold of (P(D), W\_2) which generalizes the case of Gaussian
measures) and show that it boils down to harmonic mappings valued in the
Riemannian manifold of symmetric matrices endowed with the distance coming from
optimal transport
Approximate reasoning for real-time probabilistic processes
We develop a pseudo-metric analogue of bisimulation for generalized
semi-Markov processes. The kernel of this pseudo-metric corresponds to
bisimulation; thus we have extended bisimulation for continuous-time
probabilistic processes to a much broader class of distributions than
exponential distributions. This pseudo-metric gives a useful handle on
approximate reasoning in the presence of numerical information -- such as
probabilities and time -- in the model. We give a fixed point characterization
of the pseudo-metric. This makes available coinductive reasoning principles for
reasoning about distances. We demonstrate that our approach is insensitive to
potentially ad hoc articulations of distance by showing that it is intrinsic to
an underlying uniformity. We provide a logical characterization of this
uniformity using a real-valued modal logic. We show that several quantitative
properties of interest are continuous with respect to the pseudo-metric. Thus,
if two processes are metrically close, then observable quantitative properties
of interest are indeed close.Comment: Preliminary version appeared in QEST 0
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