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Dimension-free regimes for unadjusted Langevin in ridge-separable models: A subspace robust Wasserstein analysis

Yuansi Chen, Francesco Pedrotti, Peter A. Whalley · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.31431 · cited by 0 works (OpenAlex) · open-access PDF

Spanish

El algoritmo de Langevin no ajustado (ULA) se utiliza ampliamente para la inferencia bayesiana de alta dimensión. Los análisis clásicos en la distancia de Wasserstein-2 predicen un bias asimptótico y un tiempo de mezcla que crece con la dimensión, pero esta métrica es más fuerte de lo que es necesario para muchas tareas estadísticas. En cambio, analizamos el bias en la distancia de Wasserstein robusta en el subespacio, que controla el promedio y los márgenes arbitrarios de baja dimensión y por lo tanto es adecuado para los observables típicamente estimados en la práctica. Para los potenciales separables de las colinas formados con datos i.i.d., demostramos que los límites de alta probabilidad muestran un perfil de mezcla en forma de campana, ya que el número de puntos de datos varía

Machine translation (Meta M2M100); the English original follows.

Abstract

The Unadjusted Langevin Algorithm (ULA) is widely used for high-dimensional Bayesian inference. Classical analyses in Wasserstein-2 distance predict an asymptotic bias and a mixing time that grow with dimension, but this metric is stronger than necessary for many statistical tasks. We instead analyze the bias in subspace robust Wasserstein distance, which controls the mean and arbitrary low-dimensional marginals and is therefore well suited to observables typically estimated in practice. For ridge-separable potentials formed with i.i.d. data, we prove high-probability bounds showing a bell-shaped mixing profile as the number of data points $N$ varies relative to the dimension $d$. In the overparameterized regime, the effective dimension is the rank $N$ of the data-dependent perturbation, leading to complexity depending on $N$ rather than $d$. In the large-sample regime, Gaussian-perturbation estimates yield dimension-free complexity once $N$ is sufficiently large. In contrast, near the proportional regime $\asymp d$, we prove that dimension-free guarantees are impossible in general.

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Topics: Mathematics, Mixing (physics), Wasserstein metric, Subspace topology, Dimension (graph theory), Metric (unit)

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