Low-rank Optimal Transport: Approximation, Statistics and Debiasing

@article{Scetbon2022LowrankOT,
  title={Low-rank Optimal Transport: Approximation, Statistics and Debiasing},
  author={Meyer Scetbon and Marco Cuturi},
  journal={ArXiv},
  year={2022},
  volume={abs/2205.12365}
}
The matching principles behind optimal transport (OT) play an increasingly important role in machine learning, a trend which can be observed when OT is used to disambiguate datasets in applications (e.g. single-cell genomics) or used to improve more complex methods (e.g. balanced attention in transformers or self-supervised learning). To scale to more challenging problems, there is a growing consensus that OT requires solvers that can operate on millions, not thousands, of points. The low-rank… 

Figures from this paper

References

SHOWING 1-10 OF 30 REFERENCES
Entropic regularization of continuous optimal transport problems
Extensions to McDiarmid's inequality when dierences are bounded with high probability
The method of independent bounded differences (McDiarmid, 1989) gives largedeviation concentration bounds for multivariate functions in terms of the maximum effect that changing one coordinate of the
Scikit-learn: Machine Learning in Python
Scikit-learn is a Python module integrating a wide range of state-of-the-art machine learning algorithms for medium-scale supervised and unsupervised problems. This package focuses on bringing
Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs
TLDR
This work shows how a recent variant of the OT problem that restricts the set of admissible couplings to those having a low-rank factorization is remarkably well suited to the resolution of GW, and shows that this approach is not only able to compute a stationary point of the GW problem in time O ( n 2 ) , but also uniquely positioned to benefit from the knowledge that the initial cost matrices are low- Rank.
Optimal Transport for Applied Mathematicians
Sample Complexity of Sinkhorn Divergences
TLDR
A bound is derived on the approximation error made with SDs when approximating OT as a function of the regularizer $\varepsilon", it is proved that the optimizers of regularized OT are bounded in a Sobolev (RKHS) ball independent of the two measures and the first sample complexity bound for SDs is provided.
Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein
TLDR
OTT-JAX is a python toolbox that can solve optimal transport problems between point clouds and histograms and builds on various JAX features, such as automatic and custom reverse mode differentiation, vectorization, just-in-time compilation and accelerators support.
Approximating Optimal Transport via Low-rank and Sparse Factorization
TLDR
A novel approximation for OT is proposed, in which the transport plan can be decomposed into the sum of a low-rank matrix and a sparse one, and an augmented Lagrangian method is designed to efficiently calculate the transportPlan.
Stochastic Mirror Descent: Convergence Analysis and Adaptive Variants via the Mirror Stochastic Polyak Stepsize
TLDR
New convergence guarantees for SMD with a constant stepsize are provided and a new adaptive stepsize scheme — the mirror stochastic Polyak stepsize (mSPS) is proposed — remains both practical and efficient for modern machine learning applications while inheriting the benefits of mirror descent.
Faster Wasserstein Distance Estimation with the Sinkhorn Divergence
TLDR
An estimator based on Richardson extrapolation of the Sinkhorn divergence is proposed which enjoys improved statistical and computational efficiency guarantees, under a condition on the regularity of the approximation error, which is in particular satisfied for Gaussian densities.
...
...