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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

Indonesian

The Unadjusted Langevin Algorithm (ULA) digunakan secara luas untuk inferensi Bayesian dimensi tinggi. Analisis klasik dalam jarak Wasserstein-2 memprediksi bias asymptotic dan waktu campuran yang tumbuh dengan dimensi, tetapi metrik ini lebih kuat daripada yang diperlukan untuk banyak tugas statistik. Sebaliknya, kami menganalisis bias dalam jarak Wasserstein sub-spasial yang kuat, yang mengendalikan rata-rata dan marginal dimensi rendah yang arbitrase dan karena itu cocok dengan pengamatan yang biasanya dianggarkan dalam praktek. Untuk potensial pemisahan ridge yang terbentuk dengan data i.i.d., kami membuktikan batas probabilitas tinggi yang menunjukkan profil campuran berbentuk gelombang karena jumlah titik data bervariasi dari $N$ ke dimensi $d$. Dalam rezim overparameterized, dimensi efektif adalah peringkat $N$ dari perturb

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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