Pushforward dynamics on Wasserstein spaces and measure rigidity

What the math gives to ML

The paper develops differentiable pushforward dynamics on Wasserstein space. At an invariant measure, the derivative is a transfer operator acting on perturbation vector fields followed by orthogonal projection onto the Wasserstein tangent space, and fixed tangent vectors represent first-order invariant deformations. This provides a transferable diagnostic and regularizer for recurrent neural networks and world models: estimate how hidden-state distribution perturbations are transported, then penalize gains above the stability boundary. The key falsifiable signature is a transition near spectral radius one, with perturbation norms decaying or growing geometrically on the corresponding side.