Given a source and target probability measure supported by Rd<annotation encoding="application/x-tex”>\mathbb{R}^dRdMonge's problem points to the most efficient way to map one distribution to the other. This efficiency is quantified by defining a cost function between the source and destination data. Such cost is often defaulted in the machine learning literature to the squared Euclidean distance, ℓ22(unknown,and)=12∥unknown−and∥22<annotation encoding="application/x-tex”>\ell^2_2(x,y)=\tfrac12\|xy\|_2^2ℓ22(unknown,and)=21∥unknown−and∥22. The benefits of using elastic costs, defined through a regularizer t<annotation encoding="application/x-tex”>\ cant as do(unknown,and)=ℓ22(unknown,and)+t(unknown−and)<annotation encoding="application/x-tex”>c(x, y)=\ell^2_2(x,y)+\tau(xy)do(unknown,and)=ℓ22(unknown,and)+t(unknown−and)was recently highlighted in (Cuturi et al. 2023). These costs shape the displacements of Monge's maps. t<annotation encoding="application/x-tex”>ttthat is, the difference between a source point and its image. t(unknown)−unknown<annotation encoding="application/x-tex”>T(x)-xt(unknown)−unknowngiving them a structure that coincides with that of the proximal operator of t<annotation encoding="application/x-tex”>\ cant. In this work, we make two important contributions to the study of elastic costs: (i) For any elastic cost, we propose a numerical method to compute Monge maps that are provably optimal. This provides a much-needed routine for creating synthetic problems in which the ground-truth OT map is known, by analogy with Brenier's theorem, which states that the gradient of any convex potential is always a valid Monge map for the ℓ22<annotation encoding="application/x-tex”>\ell_2^2ℓ22 cost; (ii) We propose a loss to learn the parameter. Yo<annotation encoding="application/x-tex”>\thetaYo of a parameterized regularizer tYo<annotation encoding="application/x-tex”>\tau_\thetatYoand apply it in the case where tTO(z)=∥TO⊥z∥22<annotation encoding="application/x-tex”>\tau_{A}(z)=\|A^\perp z\|^2_2tTO(z)=∥TO⊥z∥22. This regularizer promotes displacements that are in a low-dimensional subspace of Rd<annotation encoding="application/x-tex”>\mathbb{R}^dRdcovered by the p<annotation encoding="application/x-tex”>pp rows of TO∈Rp×d<annotation encoding="application/x-tex”>A\in\mathbb{R}^{p\times d}TO∈Rp×d. We illustrate the success of our procedure with synthetic data, generated using our first contribution, in which we show near-perfect recovery of TO<annotation encoding="application/x-tex”>TOTOThe subspace uses only samples. We demonstrate the applicability of this method by showing predictive improvements on single-cell data tasks.
