We analyze Reflected UAS routing for heterogeneous multi-server queues at fixed parameters under subcritical load. The deterministic surrogate is a reflected ODE on the nonnegative orthant, not the unconstrained drift equation. This reflected ODE has a unique boundary equilibrium characterized by a scalar consistency equation and a convex-potential representation; all trajectories converge to it. The older argument lifting deterministic Lyapunov descent to CTMC stability fails: the exact generator applied to the deterministic potential produces a boundary term absent from the reflected-ODE descent identity. We give a direct Foster-Lyapunov drift inequality for the CTMC using a weighted-quadratic function, bypassing the failed lift. At the benchmark parameter point, the boundary equilibrium matches the numerical attractor to machine precision, and the default Reflected UAS policy has lower mean queue length than UAS and JSSQ across independent seed blocks.
We study a solvable high-dimensional model of generative adversarial network (GAN) training in which a linear generator learns a low-dimensional subspace from data with structured latent covariance. Prior solvable GAN analyses assume unconditional signals with diagonal latent covariance; we extend the multi-feature discriminator setting to class-dependent, correlated, and non-zero-mean latent structure. For the quadratic energy discriminator, all such heterogeneity enters the dynamics through a probability-weighted effective second moment. We prove that the stochastic microscopic training process converges, in the high-dimensional limit, to deterministic ordinary differential equations governed by this effective covariance. In the matched-covariance specialization, the stability analysis yields a mode-wise solvable interval determined by the learning rates and noise level: learning begins when the leading effective eigenvalue crosses the lower threshold, while full recovery requires all relevant effective modes to remain within the interval. This reveals a signal-boosting mechanism: low-rank correlations can lift weak directions above the learnability threshold, whereas overly strong correlations destabilize recovery. Numerical simulations validate the ODE, phase boundary, and boosting mechanism. Experiments on MNIST, FashionMNIST, and CIFAR-10 further show that informed generator covariance improves alignment with the data-driven reference subspace.
We prove that the Kim--Milman flow map enjoys favorable stability properties with respect to variations in the target measure, provided that one of the target measures is sufficiently regular. Our results include stability in relative entropy, and more notably, Lipschitz stability in the $2$-Wasserstein distance up to a logarithmic factor. We complement our results with a general existence theorem for these maps for any target measure with finite second moment.