We consider the following interacting particle system in $\mathbb{R}^d$:

$$
\begin{aligned}
\dot{x}_i(t) &= -\frac{1}{N}\sum _{j=1}^{N}\nabla K(x_i(t)-x_j(t)) -\frac{1}{N} \sum _{j=1} ^{N}K\big(x_i(t)-x_j(t)\bigr)\nabla V (x_j(t)), \\\
x_i(0) &= x_i ^0 \in \mathbb R^d, \qquad i=1,\cdots,N.
\end{aligned} \tag{1}
$$

We refer to each of the $N$ functions $x_i(\cdot) \in \mathbb{R}^d$ as a particle. The function $K : \mathbb{R}^d \mapsto \mathbb{R}$ is a smooth, symmetric, and positive definite kernel. The function $V : \mathbb{R}^d \to \mathbb{R}$ is a smooth potential such that $e^{-V(x)}$ is integrable. More specific assumptions about $K$ and $V$ are given below.

We are interested in the macroscopic behavior of the particle system $(1)$ as $N \to \infty$ in the framework of mean field limit. Formally this mean field limit is described by the following non-local, nonlinear partial differential equation (PDE):
$$
\begin{aligned}
&\partial_t \rho = \nabla \cdot \left(\rho\left(K \ast (\nabla \rho+\nabla V\rho)\right)\right), \\\
&\rho(0,\cdot) = \rho_0(\cdot).
\end{aligned} \tag{2}
$$

In this article, we will prove the well-posedness of (1) and (2).

The Stein Variational Gradient Descent (SVGD) algorithm was first introduced by Liu and Wang [LW16], whose idea is to transport a set of $N$ particles $\{x_i\}_{i=1}^N$ in $\mathbb R^d$ so that their empirical measure

$$
\mu^N:=\frac{1}{N}\sum_{i=1}^N\delta_{x_i}
$$

approximates the target probability measure

$$
\rho_\infty (x)\ dx=Z^{-1} e^{-V(x)}\ dx
$$

with an unknown normalization factor $Z$.

This is the talk I presented in the Optimization Group Seminar, titled: An Introduction to Optimization over the Space of Probability Measures: From Sampling to Wasserstein Gradient Flow.

Introduction

Markov Chain

Let $(\Omega,\mathcal{F})$ be a measurable space, $P:\Omega\times \mathcal{F}\to [0,1]$ be a transition kernel, that is,

• for every $x\in \Omega$, $P(x,\cdot)$ is a probability measure on $\mathcal{F}$;
• for every $A\in \mathcal{F}$, $x\mapsto P(x,A)$ is measurable.

In generative modeling, we are given a collection of training samples $\{x_i\}_{i=1}^N$ and wish to generate new samples from the underlying target distribution $\pi$. There are already many established approaches to this problem, including likelihood-based methods, implicit generative models such as GANs, and score-based diffusion models. More recently, the flow matching framework has emerged as another powerful paradigm. In what follows, we introduce the basic ideas of flow matching and explain how works.

This is the talk I presented in the group seminar, titled: Sampling and Diffusion Model.

This is the talk I presented in the group seminar, titled: Analogy between Sampling and Optimization.