English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Computing optimal transport (OT) between measures in high dimensions isdoomed by the curse of dimensionality. A popular approach to avoid this curseis to project input measures on lower-dimensional subspaces (1D lines in thecase of sliced Wasserstein distances), solve the OT problem between thesereduced measures, and settle for the Wasserstein distance between thesereductions, rather than that between the original measures. This approach ishowever difficult to extend to the case in which one wants to compute an OT map(a Monge map) between the original measures. Since computations are carried outon lower-dimensional projections, classical map estimation techniques can onlyproduce maps operating in these reduced dimensions. We propose in this work twomethods to extrapolate, from an transport map that is optimal on a subspace,one that is nearly optimal in the entire space. We prove that the best optimaltransport plan that takes such "subspace detours" is a generalization of theKnothe-Rosenblatt transport. We show that these plans can be explicitlyformulated when comparing Gaussian measures (between which the Wassersteindistance is commonly referred to as the Bures or Fr\'echet distance). Weprovide an algorithm to select optimal subspaces given pairs of Gaussianmeasures, and study scenarios in which that mediating subspace can be selectedusing prior information. We consider applications to semantic mediation betweenelliptic word embeddings and domain adaptation with Gaussian mixture models.

Subspace Detours: Building Transport Plans that are Optimal on Subspace Projections