We study the mixing time of Metropolis-Adjusted Langevin algorithm (MALA) for
sampling a target density on $\mathbb{R}^d$. We assume that the target density
satisfies $\psi_\mu$-isoperimetry and that the operator norm and trace of its
Hessian are bounded by $L$ and $\Upsilon$ respectively. Our main result
establishes that, from a warm start, to achieve $\epsilon$-total variation
distance to the target density, MALA mixes in
$O\left(\frac{(L\Upsilon)^{\frac12}}{\psi_\mu^2}
\log\left(\frac{1}{\epsilon}\right)\right)$ iterations. Notably, this result
holds beyond the log-concave sampling setting and the mixing time depends on
only $\Upsilon$ rather than its upper bound $L d$. In the $m$-strongly
logconcave and $L$-log-smooth sampling setting, our bound recovers the previous
minimax mixing bound of MALA~\cite{wu2021minimax}.

本研究研究了在 $\mathbb {R}^d$ 上采样目标密度的 Metropolis-Adjusted Langevin 算法的混合时间，发现它在一定条件下的混合时间为 $O ((L\Upsilon)^{\frac12}/\psi_\mu^2\log (1/\epsilon))$。

关于光滑性和等周性下 Metropolis-Adjusted Langevin 算法混合的简单证明

A Simple Proof of the Mixing of Metropolis-Adjusted Langevin Algorithm  under Smoothness and Isoperimetry

Pedestrian detection in a crowd is a very challenging issue. This paper
addresses this problem by a novel Non-Maximum Suppression (NMS) algorithm to
better refine the bounding boxes given by detectors. The contributions are
threefold: (1) we propose adaptive-NMS, which applies a dynamic suppression
threshold to an instance, according to the target density; (2) we design an
efficient subnetwork to learn density scores, which can be conveniently
embedded into both the single-stage and two-stage detectors; and (3) we achieve
state of the art results on the CityPersons and CrowdHuman benchmarks.

本文提出一种使用新型非极大值抑制（NMS）算法解决人群中的行人检测问题，使用自适应 NMS 和学习密度得分的子网络，可以有效改善边界框的精确度，并在 CityPersons 和 CrowdHuman 基准测试中取得了最先进的结果。

适应性 NMS: 在人群中优化行人检测

Adaptive NMS: Refining Pedestrian Detection in a Crowd

The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm
which makes local moves by incorporating information about the gradient of the
logarithm of the target density. In this paper we study the efficiency of MALA
on a natural class of target measures supported on an infinite dimensional
Hilbert space. These natural measures have density with respect to a Gaussian
random field measure and arise in many applications such as Bayesian
nonparametric statistics and the theory of conditioned diffusions. We prove
that, started in stationarity, a suitably interpolated and scaled version of
the Markov chain corresponding to MALA converges to an infinite dimensional
diffusion process. Our results imply that, in stationarity, the MALA algorithm
applied to an N-dimensional approximation of the target will take
$\mathcal{O}(N^{1/3})$ steps to explore the invariant measure, comparing
favorably with the Random Walk Metropolis which was recently shown to require
$\mathcal{O}(N)$ steps when applied to the same class of problems.

本文研究了 Metropolis-adjusted Langevin (MALA) 算法在基于无限维 Hibert 空间的自然目标测度上的效率，证明了该算法对于该类问题的收敛速度比 Random Walk Metropolis 更快，并可以探索不变量的度量，所需步数是其 $N$ 维逼近的第 $1/3$ 倍，适用于诸如 Bayesian 非参数统计学和条件扩散理论等领域。