2021/01/17 by David Jekel, Jekel, David, Wuchen Li +3 · 4 citations
Computer Science · Mathematics · Medicine · Physics and Astronomy · #35Q49 #46L52 #46L54 #94A17 #Advanced Neuroimaging Techniques and Applications #FOS: Computer and information sciences #FOS: Mathematics #Geometric Analysis and Curvature Flows #Information Theory (cs.IT) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #cs.IT #math.IT #math.OA #msc:35Q49 #msc:46L52 #msc:46L54 #msc:94A17
paper · pdf · doi:10.48550/arxiv.2101.06572
134 pages, revised
openalex publication_date 2021/01/17 · arxiv created 2021/10/25 · arxiv updated 2021/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We formulate a free probabilistic analog of the Wasserstein manifold on ℝd (the formal Riemannian manifold of smooth probability densities on ℝd), and we use it to study smooth non-commutative transport of measure. The points of the free Wasserstein manifold \mathscrW(ℝ*d) are smooth tracial non-commutative functions V with quadratic growth at ∞, which correspond to minus the log-density in the classical setting. The space of smooth tracial non-commutative functions used here is a new one whose definition and basic properties we develop in the paper; they are scalar-valued functions of self-adjoint d-tuples from arbitrary tracial von Neumann algebras that can be approximated by trace polynomials. The space of non-commutative diffeomorphisms \mathscrD(ℝ*d) acts on \mathscrW(ℝ*d) by transport, and the basic relationship between tangent vectors for \mathscrD(ℝ*d) and tangent vectors for \mathscrW(ℝ*d) is described using the Laplacian LV associated to V and its pseudo-inverse ΨV (when defined). Following similar arguments to arXiv:1204.2182, arXiv:1701.00132, and arXiv:1906.10051 in the new setting, we give a rigorous proof for the existence of smooth transport along any path t ↦ Vt when V is sufficiently close (1/2) ∑j tr(xj2), as well as smooth triangular transport.