Sliced Optimal Transport on Real Projective Space

A projective space is the set of straight lines going through the origin in a vector space. This corresponds to the set of undirected orientations. Although it has been little studied in machine learning (ML), it implicitly appears in many problems, starting with eigendecomposition. In practice, such orientations are often represented with unit vectors, i.e. points on the sphere, that may nevertheless be very far apart from one another even though the orientations are similar. This may lead to discontinuities and misleading notions of distance and averaging. While optimal transport enables the comparison of probability measures on any metric space, coming with many applications in ML, its computation is expensive. We introduce a sliced Wasserstein distance on real projective spaces, denoted $PSW$, which enables efficient comparison of probability distributions on these manifolds. It projects the projective measures on geodesics and computes the average of the OT costs obtained on those circular geodesics. We experimentally illustrate the importance of considering the projective framework, when it holds, rather than the spherical one, and we give a glimpse on possible applications.