1,570 research outputs found
Learning Generative Models across Incomparable Spaces
Generative Adversarial Networks have shown remarkable success in learning a
distribution that faithfully recovers a reference distribution in its entirety.
However, in some cases, we may want to only learn some aspects (e.g., cluster
or manifold structure), while modifying others (e.g., style, orientation or
dimension). In this work, we propose an approach to learn generative models
across such incomparable spaces, and demonstrate how to steer the learned
distribution towards target properties. A key component of our model is the
Gromov-Wasserstein distance, a notion of discrepancy that compares
distributions relationally rather than absolutely. While this framework
subsumes current generative models in identically reproducing distributions,
its inherent flexibility allows application to tasks in manifold learning,
relational learning and cross-domain learning.Comment: International Conference on Machine Learning (ICML
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
Markovian Sliced Wasserstein Distances: Beyond Independent Projections
Sliced Wasserstein (SW) distance suffers from redundant projections due to
independent uniform random projecting directions. To partially overcome the
issue, max K sliced Wasserstein (Max-K-SW) distance (), seeks the best
discriminative orthogonal projecting directions. Despite being able to reduce
the number of projections, the metricity of Max-K-SW cannot be guaranteed in
practice due to the non-optimality of the optimization. Moreover, the
orthogonality constraint is also computationally expensive and might not be
effective. To address the problem, we introduce a new family of SW distances,
named Markovian sliced Wasserstein (MSW) distance, which imposes a first-order
Markov structure on projecting directions. We discuss various members of MSW by
specifying the Markov structure including the prior distribution, the
transition distribution, and the burning and thinning technique. Moreover, we
investigate the theoretical properties of MSW including topological properties
(metricity, weak convergence, and connection to other distances), statistical
properties (sample complexity, and Monte Carlo estimation error), and
computational properties (computational complexity and memory complexity).
Finally, we compare MSW distances with previous SW variants in various
applications such as gradient flows, color transfer, and deep generative
modeling to demonstrate the favorable performance of MSW.Comment: Accepted to NeurIPS 2023, 29 pages, 8 figures, 5 table
Fast Optimal Transport Averaging of Neuroimaging Data
Knowing how the Human brain is anatomically and functionally organized at the
level of a group of healthy individuals or patients is the primary goal of
neuroimaging research. Yet computing an average of brain imaging data defined
over a voxel grid or a triangulation remains a challenge. Data are large, the
geometry of the brain is complex and the between subjects variability leads to
spatially or temporally non-overlapping effects of interest. To address the
problem of variability, data are commonly smoothed before group linear
averaging. In this work we build on ideas originally introduced by Kantorovich
to propose a new algorithm that can average efficiently non-normalized data
defined over arbitrary discrete domains using transportation metrics. We show
how Kantorovich means can be linked to Wasserstein barycenters in order to take
advantage of an entropic smoothing approach. It leads to a smooth convex
optimization problem and an algorithm with strong convergence guarantees. We
illustrate the versatility of this tool and its empirical behavior on
functional neuroimaging data, functional MRI and magnetoencephalography (MEG)
source estimates, defined on voxel grids and triangulations of the folded
cortical surface.Comment: Information Processing in Medical Imaging (IPMI), Jun 2015, Isle of
Skye, United Kingdom. Springer, 201
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