Let $M$ be a smooth manifold and $p \in M$. The tangent space at $p$, denoted $T_p M$, is defined as the space of derivations at $p$, i.e., linear maps $$ v : C^\infty(M) \to \mathbb{R} $$ satisfying the Leibniz rule: $$ v(fg) = … Read More →

Pushforward (for tangent vectors) Let $v \in T_p M$ be a tangent vector at point $p \in M$. Then: $$ d \varphi_p: T_p M \rightarrow T_{\varphi(p)} N $$ is the pushforward map (also called the differential of $\varphi$ ). $$ … Read More →

Under strong logconcavity, the Wasserstein distance between the biased limit of ULA and $\nu$ is bounded by terms proportional to $\sqrt{\epsilon}$, with $\epsilon$ being the discretization parameter. […] If the covariances … Read More →

For a smooth function $f:M\to\mathbb{R}$, define $$ \Gamma(f) ;:=; |\nabla f|^2. $$ The operator $\Gamma_2$ is given by $$ \Gamma_2(f) ;:=; \frac12,\bigl(\mathcal{L}\Gamma(f) - 2,\Gamma(f),\mathcal{L} f\bigr), $$ where … Read More →

is a smooth manifold that sits inside some ambient euclidean space $\mathbb{R}^n$, with the subspace topology and differential structure inherited from $\mathbb{R}^n$. It is the smooth injective immersion which is also a … Read More →

The SDE for the underdamped Langevin dynamics is modified to incorporate anisotropic smoothness through a position-dependent metric tensor: $$ dX_t = -G(x)^{-1} \nabla U(x) , dt + \sqrt{2G(x)^{-1}} , dB_t, $$ where $G(x) \in … Read More →

sinho chewi’s book gradient flows ambrosio optimal transport villani (not topics) Read More →

\cite{dalalyan2017} and \cite{durmus_non-asymptotic_2016} established error bounds for ULA based on step size constraints. \cite{eberle_couplings_2017} reflection and synchronous couplings to quantify contraction rates in … Read More →

Tweedie deals with conditions under which the convergence of langevin diffusion is exponentially fast, and whether it extends to higher moments. The Unadjusted Langevin Algorithm (ULA) is a discrete-time Markov chain $\mathbf{U}n$ … Read More →

The Langevin diffusion and the Unadjusted Langevin Algorithm (ULA) can be interpreted as gradient flow dynamics minimizing the relative entropy (KL divergence) over the space of probability measures \cite{jordan_variational_1998}. … Read More →

A set $C \in \mathcal{B}(\mathrm{X})$ is called a small set if there exists an $m>0$, and a non-trivial measure $\nu_m$ on $\mathcal{B}(\mathrm{X})$, such that for all $x \in C, B \in \mathcal{B}(\mathrm{X})$, $$ P^m(x, B) \geq … Read More →

The probability density function $\rho_t$ of $X_t$ evolves according to the Fokker-Planck equation: $$ \frac{\partial \rho_t}{\partial t} = \nabla \cdot \left(\rho_t \nabla \log \frac{\rho_t}{\nu}\right) = \nabla \cdot (\rho_t … Read More →

\cite{cheng_underdamped_2018} studied ULMC as a variant of HMC algorithm, which converges to $\varepsilon$ error in 2-Wasserstein distance after $\mathcal{O}\left(\frac{\sqrt{d} \kappa^2}{\varepsilon}\right)$ iterations, under the … Read More →

In discrete time, we usually want to use a step size $\eta$ to construct an iterative update rule for the discretization. […] Unadjusted Langevin Algorithm (ULA) is a simple discretization of the Langevin diffusion: $$ … Read More →

We note that the logarithmic sobolev inequality (LSI) provides a gradient domination condition that ensures exponential convergence of langevin dynamics under the wasserstein metric\cite{otto2000generalization}. $$ … Read More →

Langevin dynamics connects probability theory and optimization through its formulation as a gradient flow in the Wasserstein-2 space. The time evolution of the Fokker-Planck equation\cite{jordan_variational_1998} can be … Read More →

The goal is to efficiently sample from a target distribution $\pi \propto e^{-f(x)}$, in high dimensional or non-convex settings, direct sampling difficult and usually we define a stochastic process whose stationary distribution … Read More →

the point of this exposition is thinking about the log-sobolev inequality in situations where you have high probability guarantees that the conditions for LSI are met. does this mean exponential convergence is still guaranteed? … Read More →

this webpage serves as a dump of my personal notes on sampling, optimization and algorithms. these are always a work in progress, feel free to email me with corrections or ideas, but please note that these are not maintained for … Read More →