Ali Baheri's new paper tightens Wasserstein bounds for nonlinear systems
A novel distributional bound that outperforms classical conservative estimates in contracting flows.
Ali Baheri’s latest preprint tackles a fundamental limitation in contraction theory: the classical Wasserstein bound on distributional convergence is tight only for linear systems. For nonlinear contracting dynamics, the bound collapses the spatially varying local contraction rate into a single worst-case constant, discarding valuable distributional information. Baheri introduces three concrete improvements. First, he derives a new Wasserstein bound that replaces the worst-case rate with a displacement-weighted distributional average of the local contraction rate, strictly improving the classical bound for every nonlinear contracting system. Second, the paper provides the first theoretical characterization of Euler discretization error under contraction: the error profile is non-monotone, peaks at a universal time depending only on the contraction rate, then decays exponentially—a behavior unique to contracting dynamics. Third, Baheri proves that nonlinear contracting drifts always achieve strictly smaller stationary variance than a linear system sharing the same worst-case contraction rate, formally establishing a noise-rejection advantage for nonlinear controllers.
These results are validated on a suite of one- and two-dimensional vector fields. The work has direct implications for ensemble control, Bayesian estimation, and generative modeling—any domain where initial conditions are uncertain and represented as probability distributions. By tightening distributional convergence bounds, practitioners can obtain more accurate performance guarantees and error estimates for nonlinear systems. The self-correcting Euler error insight is particularly relevant for numerical simulations, where discretization error is often assumed to accumulate monotonically. The noise tightening property further suggests that nonlinear controllers can outperform linear ones in rejecting disturbances, a fact now supported by rigorous theory rather than empirical observation alone.
- New Wasserstein bound uses displacement-weighted average contraction rate, improving on worst-case constant for all nonlinear contracting systems.
- First theoretical proof that Euler discretization error under contraction peaks at a universal time then decays exponentially, not monotonically.
- Nonlinear contracting drifts achieve strictly smaller stationary variance than linear systems with the same worst-case contraction rate.
Why It Matters
Tighter theoretical bounds enable better performance guarantees for nonlinear control, Bayesian inference, and generative models with uncertain initial conditions.