Research map / Optimization / Nonconvex and smooth optimization
Interaction-Force Transport Gradient Flows
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Proposes interaction-force transport (IFT) gradient flows over non-negative and probability measures, combining Wasserstein and kernel MMD geometries via inf-convolution, with a JKO-splitting particle algorithm and global exponential convergence guarantees for MMD and KL energies.
Problem
Gradient flows for minimizing MMD or KL energies over measures lack global convergence guarantees; MMD-based particle flows collapse to modes or get stuck at local minima and need heuristic noise injection, and kernel-mean-embedding methods cannot transport mass.
Approach
Construct a new gradient flow dissipation geometry by inf-convolving the Wasserstein tensor with a kernelized (MMD) Hellinger-type tensor, yielding a reaction-diffusion-like flow that both transports and teleports mass; a mass-preserving spherical variant is derived for probability measures.
Key results
- Theory: IFT flow of squared MMD energy satisfies a global Lojasiewicz inequality with constant c >= 2 beta, giving exponential decay exp(-2 beta t) of the squared MMD.
- Spherical IFT flow of KL energy decays as exp(-2 alpha lambda t) when target is lambda-log-concave (Bakry-Emery/LSI).
- Numerically, the IFT particle algorithm reaches accuracy similar to noise-injected MMD flow in the 2D Gaussian experiment, and does so much faster (already after 1000 steps) without noise injection; experiments are small-scale synthetic only.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use spherical IFT instead of vanilla MMD flow | MMD sampling · Gaussian target in 2D | sample quality | Helped |
| Use spherical IFT instead of noise-injected MMD flow | MMD sampling · Gaussian target in 2D | accuracy | No clear effect |
| Use spherical IFT instead of noise-injected MMD flow | MMD sampling · Gaussian target in 2D | iterations to approach target | Helped |
| Use spherical IFT instead of noise-injected MMD flow | MMD sampling · Gaussian target in 2D | need for heuristic noise injection | Helped |
| Use IFT gradient flow for squared MMD energy | IFT gradient flow · Non-negative measures | squared MMD energy decay | Helped |
| Use spherical IFT gradient flow for KL energy | Spherical IFT gradient flow · Probability measures with a -log-concave target | KL divergence decay | Helped |
Surprising findings
- Spherical MMD gradient flow of squared MMD coincides with a linear interpolation between initial measure and target, mu_t = e^{-t} mu_0 + (1-e^{-t}) pi.
Limitations
- Experiments are small synthetic 2D and 100D Gaussian/mixture targets, not competitive benchmarks
- Global convergence guarantees do not extend to the full IFT flow over non-negative measures for KL energy (no global LSI on M+)
- Inverse of the compact integral operator is unbounded; requires viscosity regularization in practice
Experimental setup
Datasets: synthetic Gaussian target samples synthetic Gaussian mixture target samples
Benchmarks: 2D Gaussian target 2D Gaussian mixture target 100D Gaussian mixture target
Metrics: squared MMD loss
Baselines: MMD flow (Arbel et al.) MMD flow with noise injection Wasserstein-Fisher-Rao flow of MMD
Builds on: Wasserstein gradient flow Hellinger-Kantorovich (Wasserstein-Fisher-Rao) gradient flow MMD flow JKO scheme Bakry-Emery theorem
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