In recent years, Rectified flow (RF) has gained considerable popularity largely due to its generation efficiency and state-of-the-art performance. In this paper, we investigate how well RF automatically adapts to the intrinsic low dimensionality of the support of the target distribution to accelerate sampling. We show that, using a carefully designed \blue{end-point adapted} time-discretization scheme and with sufficiently accurate drift estimates, the RF sampler enjoys an iteration complexity of order $O(k/\varepsilon)$ (up to log factors), where $\varepsilon$ is the precision in total variation distance and $k$ is the intrinsic dimension of the target distribution. In addition, we show that the denoising diffusion probabilistic model (DDPM) procedure is equivalent to a stochastic version of RF by establishing a novel connection between these processes and stochastic localization. Building on this connection, we further design a stochastic RF sampler that also adapts to the low-dimensionality of the target distribution under mild requirements on the accuracy of the drift estimates, and also with a specific time schedule. We illustrate the efficacy of newly designed time-discretization schedules with simulations on the synthetic data and text-to-image (T2I) data experiments.
Sparse mixtures of experts combine multiple prediction models while evaluating only a small subset for each input. Although this architecture reduces computation, how expert separation and routing sparsity jointly govern expert recovery remains an open problem. In this paper, we address this question for location Gaussian mixtures of experts with a fixed number of experts by establishing how expert separation and spatial arrangement determine recovery rates. For dense models, our results reveal that when experts form well-separated clusters, the separation geometry is governed by the size of the largest cluster; without a prescribed cluster structure, this geometry is instead driven by the total number of experts. For sparse models, the separation geometry on retained routing regions is determined by the number of active experts, while the contribution of omitted regions is controlled by their total probability. Consequently, activating fewer experts reduces the separation complexity in these bounds. Furthermore, Top-1 routing yields a particularly striking result: experts can be recovered at a rate independent of the separation quantity. Our technical approach combines polynomial interpolation and Gaussian reproducing-kernel Hilbert space (RKHS) test functions with quantitative restriction inequalities that extend pointwise comparisons to expert functions.