Unbalanced Optimal Transport, from Theory to Numerics

Abstract

Optimal Transport (OT) has recently emerged as a central tool in data sciences to compare in a geometrically faithful way point clouds and more generally probability distributions. The wide adoption of OT into existing data analysis and machine learning pipelines is however plagued by several shortcomings. This includes its lack of robustness to outliers, its high computational costs, the need for a large number of samples in high dimension and the difficulty to handle data in distinct spaces. In this review, we detail several recently proposed approaches to mitigate these issues. We insist in particular on unbalanced OT, which compares arbitrary positive measures, not restricted to probability distributions (i.e. their total mass can vary). This generalization of OT makes it robust to outliers and missing data. The second workhorse of modern computational OT is entropic regularization, which leads to scalable algorithms while lowering the sample complexity in high dimension. The last point presented in this review is the Gromov-Wasserstein (GW) distance, which extends OT to cope with distributions belonging to different metric spaces. The main motivation for this review is to explain how unbalanced OT, entropic regularization and GW can work hand-in-hand to turn OT into efficient geometric loss functions for data sciences.

scConfluence : single-cell diagonal integration with regularized Inverse Optimal Transport on weakly connected features

“…As real data often contains outliers to which OT is highly sensitive, a more robust extension of OT called unbalanced OT 33 has been developed. where τ is a positive parameter controlling the looseness of the relaxation.…”

“…Therefore, our approach does not suffer from the loss of biological information generally resulting from modality conversion prior to dimension reduction. In addition, thanks to the unbalanced relaxation of Optimal Transport 33 , scConfluence can also deal with cell types absent in a modality thus overcoming all the major limitations of the state-of-the-art.…”

DeST-OT: Alignment of Spatiotemporal Transcriptomics Data

“…Just as ξ arises from Π, vector γ 1 is defined from T through (37) and (35). Given the output (Π, ξ) of an OT method, the next proposition gives the optimal pair (T, γ 1 ) for this output according to the growth distortion metric (36).…”

“…Note that by considering the dual problem, partial optimal transport , in which there is a constraint s ∈ (0, 1) on the total mass transported, can be seen as an instance of unbalanced optimal transport associated to the total variation φ -divergence (see [36] § 4.2). Unbalanced optimal transport refers to the version of (14) obtained by deleting the two hard constraints, and , on the marginals of Π , and instead adding to the cost function in brackets a pair of “soft constraints” in the form of two terms:.…”

Modeling Single-Cell Dynamics Using Unbalanced Parameterized Monge Maps

“…ỹn ∼ ν. Such couplings πUOT can be estimated efficiently using entropic regularization (Cuturi, 2013;Séjourné et al, 2022) (Appendix B.1). Moreover, we can learn the re-weighting functions as we have access to estimates of their pointwise evaluation:…”