  Matching a source to a target probability measure is often solved by
instantiating a linear optimal transport (OT) problem, parameterized by a
ground cost function that quantifies discrepancy between points. When these
measures live in the same metric space, the ground cost often defaults to its
distance. When instantiated across two different spaces, however, choosing that
cost in the absence of aligned data is a conundrum. As a result, practitioners
often resort to solving instead a quadratic Gromow-Wasserstein (GW) problem. We
exploit in this work a parallel between GW and cost-regularized OT, the
regularized minimization of a linear OT objective parameterized by a ground
cost. We use this cost-regularized formulation to match measures across two
different Euclidean spaces, where the cost is evaluated between transformed
source points and target points. We show that several quadratic OT problems
fall in this category, and consider enforcing structure in linear transform
(e.g. sparsity), by introducing structure-inducing regularizers. We provide a
proximal algorithm to extract such transforms from unaligned data, and
demonstrate its applicability to single-cell spatial transcriptomics/multiomics
matching tasks.
