The goal of this article is to introduce the Hamiltonian Monte Carlo (HMC)
method -- a Hamiltonian dynamics-inspired algorithm for sampling from a Gibbs
density $\pi(x) \propto e^{-f(x)}$. We focus on the "idealized" case, where one
can compute continuous trajectories exactly. We show that idealized HMC
preserves $\pi$ and we establish its convergence when $f$ is strongly convex
and smooth.

介绍了哈密尔顿蒙特卡罗方法 (HMC)—— 基于哈密尔顿动力学的一种采样算法，用于从 Gibbs 密度中采样。重点在于 “理想化” 情况，其中能够精确计算连续轨迹，表明理想化 HMC 能保持 π 并在 f 为强凸和光滑时收敛。

Hamiltonian Monte Carlo 方法采样简介

An Introduction to Hamiltonian Monte Carlo Method for Sampling

We develop a novel framework to study smooth and strongly convex optimization
algorithms, both deterministic and stochastic. Focusing on quadratic functions
we are able to examine optimization algorithms as a recursive application of
linear operators. This, in turn, reveals a powerful connection between a class
of optimization algorithms and the analytic theory of polynomials whereby new
lower and upper bounds are derived. Whereas existing lower bounds for this
setting are only valid when the dimensionality scales with the number of
iterations, our lower bound holds in the natural regime where the
dimensionality is fixed. Lastly, expressing it as an optimal solution for the
corresponding optimization problem over polynomials, as formulated by our
framework, we present a novel systematic derivation of Nesterov's well-known
Accelerated Gradient Descent method. This rather natural interpretation of AGD
contrasts with earlier ones which lacked a simple, yet solid, motivation.

我们开发了一个新的框架来研究光滑和强凸优化算法，特别是针对二次函数，我们能够将优化算法作为线性运算的递归应用程序来检查，这揭示了一种强大的联系，即一类优化算法与多项式的分析理论之间的联系，从而导出了新的下界和上界，同时我们还以多项式相关的最优解的形式表达它，从而对 Nesterov 著名的加速梯度下降方法进行了新的系统推导。

关于平滑和强凸优化问题的下限和上限

On Lower and Upper Bounds for Smooth and Strongly Convex Optimization  Problems

We complete the study of smooth $Z_3$-quotients of complete intersection
Calabi-Yau threefolds by discussing the six new manifolds that admit free $Z_3$
actions that were discovered discovered recently by Braun. These manifolds were
missed in an earlier work and complete the web of smooth $Z_3$-quotients in a
nice way. We discuss the transitions between these manifolds and include also
the other manifolds of the web. This leads to the conclusion that the web of
$Z_3$-free quotients of complete intersection Calabi-Yau threefolds is
connected by conifold transitions.

本文讨论了新发现的六种完全交叉 Calabi-Yau 三倍体，这些倍体错过了早期工作，并且可通过圆锥转换连接到交叉 Calabi-Yau 三倍体的自由 $Z_3$ 作用的其他倍体，从而表明了 $Z_3$-free 商空间的网络通过圆锥转换相连。