Classical analyses of the Unadjusted Langevin Algorithm (ULA) in Wasserstein-2 distance predict a dimension-dependent mixing time and bias, yet empirically ULA often estimates low-dimensional statistical functionals accurately without scaling the step size with dimension. This paper proposes using the subspace robust Wasserstein (SRW) distance to analyze ULA on ridge-separable Bayesian posteriors, proving that iteration complexity can be dimension-free in both the overparameterized regime ($N \lesssim d$) and the large-sample regime ($N \gtrsim d^2$), while showing such guarantees are impossible near the proportional regime $N \asymp d$. The results explain a bell-shaped bias profile and provide a rotationally invariant alternative to prior $\ell^\infty$-based dimension-free analyses that fail under isotropy.

Score-based generative models with heavy-tailed $\alpha$-stable Lévy noise have shown empirical promise for extreme-event modeling, yet theoretical guarantees remain scarce. This paper develops a non-asymptotic $W_p$ convergence framework for scale-preserving $\alpha$-stable dynamics and proves that global Wasserstein convergence alone does not separate stable from Brownian models on heavy-tailed targets. A sharp distinction emerges instead in the tails: stable dynamics inherit power-law tails with index $\alpha$, while Brownian outputs stay light-tailed, and asymptotically correct extreme-quantile scaling requires matching the target tail index $\nu$ to the noise index $\alpha$. These results establish tail-index matching as a principled design mechanism for capturing heavy-tailed phenomena.

The paper tackles sampling from Gibbs distributions on matrix spaces with potentials that are neither convex nor globally gradient-Lipschitz. Its core idea is to replace the quadratic kinetic energy of underdamped Langevin dynamics with a non-quadratic, relativistic-like energy $K_\delta(p)=\delta^{-1}\operatorname{Tr}[(I_n+\delta pp^\top)^{1/2}-I_n]$, whose gradient tames momentum singular values smoothly. By carefully coupling a momentum-dependent diffusion matrix $M_\delta(p)$ and its divergence correction $\Gamma_\delta(p)$, the authors preserve the exact target Gibbs measure, recover regularised Muon in the zero-temperature limit, and prove exponential convergence of the diffusion plus uniform-in-time moment bounds for an explicit Euler-Maruyama scheme.

Many recent sampling algorithms built on Wasserstein gradient flows (WGF) and forward-only diffusion processes (FODP) come with exponential-convergence guarantees that are often read as proof of fast practical mixing. This paper argues that guarantee is misleading: the canonical KL-induced WGF and overdamped forward diffusion produce the same marginal Fokker--Planck evolution, so they inherit the same metastability and exponentially slow inter-mode transport known in statistical physics. Using spectral-gap and mean-first-passage-time analysis on multimodal targets, the authors show that well-separated modes induce mixing times that scale exponentially with barrier height, and that standard log-linear annealing does not remove this scaling. The conclusion is that purely local, gradient-driven sampling is structurally limited for multimodal distributions, motivating nonlocal transport mechanisms.

This paper develops a neural operator framework for approximating mappings defined on constrained Wasserstein spaces $\mathcal{M}_\lambda$, consisting of probability measures on $I \times \mathbb{R}^d$ with prescribed marginal $\lambda$ on the label space $I$. The core contribution is the DeepONetCyl architecture, which combines cylindrical moment approximations $\Phi_J(\mu) = (\langle \varphi_1, \mu \rangle, \ldots, \langle \varphi_J, \mu \rangle)$ with a DeepONet-type branch–trunk structure to preserve the marginal constraint. This enables learning of heterogeneous (non-exchangeable) mean-field control problems where agent interactions depend on labels, extending prior neural methods beyond the exchangeable case.

This paper extends stochastic approximation (SA) theory to non-Markovian driving noise that is also non-ergodic, establishing that the ergodic decomposition of the original process corresponds to a Doeblin decomposition of an equivalent Markov chain. The core insight is that iterates retain memory of the distant past through the tail $\sigma$-field at $-\infty$, offering a theoretical lens on how learning algorithms might encode long-term dependencies. The author proposes this framework as a paradigm for understanding transformer attention mechanisms and continual learning, where the entire history influences current updates.