65,593 research outputs found
Inverse optimal transport
Discrete optimal transportation problems arise in various contexts in
engineering, the sciences and the social sciences. Often the underlying cost
criterion is unknown, or only partly known, and the observed optimal solutions
are corrupted by noise. In this paper we propose a systematic approach to infer
unknown costs from noisy observations of optimal transportation plans. The
algorithm requires only the ability to solve the forward optimal transport
problem, which is a linear program, and to generate random numbers. It has a
Bayesian interpretation, and may also be viewed as a form of stochastic
optimization.
We illustrate the developed methodologies using the example of international
migration flows. Reported migration flow data captures (noisily) the number of
individuals moving from one country to another in a given period of time. It
can be interpreted as a noisy observation of an optimal transportation map,
with costs related to the geographical position of countries. We use a
graph-based formulation of the problem, with countries at the nodes of graphs
and non-zero weighted adjacencies only on edges between countries which share a
border. We use the proposed algorithm to estimate the weights, which represent
cost of transition, and to quantify uncertainty in these weights
Learning Generative Models with Sinkhorn Divergences
The ability to compare two degenerate probability distributions (i.e. two
probability distributions supported on two distinct low-dimensional manifolds
living in a much higher-dimensional space) is a crucial problem arising in the
estimation of generative models for high-dimensional observations such as those
arising in computer vision or natural language. It is known that optimal
transport metrics can represent a cure for this problem, since they were
specifically designed as an alternative to information divergences to handle
such problematic scenarios. Unfortunately, training generative machines using
OT raises formidable computational and statistical challenges, because of (i)
the computational burden of evaluating OT losses, (ii) the instability and lack
of smoothness of these losses, (iii) the difficulty to estimate robustly these
losses and their gradients in high dimension. This paper presents the first
tractable computational method to train large scale generative models using an
optimal transport loss, and tackles these three issues by relying on two key
ideas: (a) entropic smoothing, which turns the original OT loss into one that
can be computed using Sinkhorn fixed point iterations; (b) algorithmic
(automatic) differentiation of these iterations. These two approximations
result in a robust and differentiable approximation of the OT loss with
streamlined GPU execution. Entropic smoothing generates a family of losses
interpolating between Wasserstein (OT) and Maximum Mean Discrepancy (MMD), thus
allowing to find a sweet spot leveraging the geometry of OT and the favorable
high-dimensional sample complexity of MMD which comes with unbiased gradient
estimates. The resulting computational architecture complements nicely standard
deep network generative models by a stack of extra layers implementing the loss
function
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