A Jump-Modified Wasserstein Gradient Flow for Maximum Mean Discrepancy with Energy Kernels

Key points

• Under subcritical density and finite-moment assumptions, we prove global well-posedness.1
• Under our hypotheses on the target and jump rates, every Borel probability measure with finite MMD energy and zero jump dissipation is equal to the target.1
• We also construct a global collision-free particle process whose energy is nonincreasing along each path.1
• A quantitative mean-field estimate applies to general initial empirical approximations, including deterministic configurations; propagation of chaos for exchangeable data and long-time target approximation follow as consequences.1

Sources (1)

• [1]A Jump-Modified Wasserstein Gradient Flow for Maximum Mean Discrepancy with Energy Kernels
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 21, 12:16 AM

  We study a jump-modified Wasserstein gradient flow of the squared maximum mean discrepancy generated by the energy kernels $K(z)=-|z|^q$, $0<q<2$, in every dimension $d\ge2$ as well as the one-dimensional range $1\le q<2$.

  Under subcritical density and finite-moment assumptions, we prove global well-posedness.

Extractive summary: sentences quoted from the sources.