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‎doc/src/week15/Latexfiles/compare.tex‎

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\documentclass{beamer}
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\usepackage{amsmath,amsfonts,amssymb,bm}
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\usepackage{physics}
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\usetheme{Madrid}
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\title{Diffusion Models and Normalizing Flows}
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\author{MHJ}
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\date{March 29, 2025}
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\begin{document}
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\frame{\titlepage}
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%================================================
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\section{Motivation}
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%================================================
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\begin{frame}{Generative Modeling}
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Goal: learn a distribution
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\[
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p(x), \quad x \in \mathbb{R}^d
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\]
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Two major paradigms:
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\begin{itemize}
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\item Normalizing flows (deterministic)
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\item Diffusion models (stochastic)
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\end{itemize}
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\end{frame}
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%================================================
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\section{Normalizing Flows}
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%================================================
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\begin{frame}{Basic Idea}
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\[
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z \sim p_0(z), \quad x = f(z)
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\]
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\[
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p(x) = p_0(z)\left|\det \frac{\partial z}{\partial x}\right|
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Change of Variables}
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\[
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\log p(x) = \log p_0(z) - \log |\det J|
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Composition}
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\[
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x = f_K \circ \cdots \circ f_1(z)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Continuous Flow}
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\[
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\frac{dx}{dt} = v(x,t)
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\]
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\[
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\frac{d}{dt}\log p(x) = -\nabla \cdot v
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Interpretation}
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\begin{itemize}
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\item deterministic transport
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\item invertible mapping
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\end{itemize}
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\end{frame}
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%================================================
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\section{Diffusion Models}
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%================================================
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\begin{frame}{Forward Process}
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\[
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dx = -\frac{1}{2}\beta(t)x\,dt + \sqrt{\beta(t)}\,dW_t
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Fokker–Planck Equation}
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\[
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\partial_t p = -\nabla \cdot (f p) + \frac{1}{2}\nabla^2 (\sigma^2 p)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Reverse Process}
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\[
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dx = [f(x,t) - \sigma^2 \nabla \log p(x,t)]dt + \sigma d\bar{W}_t
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Score Function}
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\[
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s(x,t) = \nabla \log p(x,t)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Training Objective}
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\[
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\mathbb{E}\|s_\theta(x,t) - \nabla \log p(x,t)\|^2
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\]
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\end{frame}
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%================================================
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\section{Connection Between the Two}
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%================================================
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\begin{frame}{Common Structure}
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Both aim to construct:
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\[
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T: p_0 \rightarrow p(x)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Flows vs Diffusion}
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Flows:
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\[
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x = f(z)
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\]
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Diffusion:
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\[
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dx = v(x,t)dt + \sigma dW_t
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Deterministic vs Stochastic}
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\begin{itemize}
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\item flows: deterministic transport
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\item diffusion: stochastic evolution
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\end{itemize}
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\end{frame}
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%================================================
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\section{PDE Perspective}
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%================================================
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\begin{frame}{Continuity Equation (Flows)}
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\[
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\partial_t p + \nabla \cdot (v p) = 0
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Fokker–Planck (Diffusion)}
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\[
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\partial_t p = -\nabla \cdot (v p) + \frac{1}{2}\nabla^2 (\sigma^2 p)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Key Difference}
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\begin{itemize}
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\item flows: transport only
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\item diffusion: transport + noise
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\end{itemize}
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\end{frame}
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%================================================
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\section{Statistical Physics Interpretation}
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%================================================
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\begin{frame}{Boltzmann Distribution}
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\[
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p(x) \propto e^{-E(x)}
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Diffusion as Langevin Dynamics}
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\[
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dx = -\nabla E(x)dt + \sqrt{2}dW_t
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Flows as Deterministic Maps}
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\begin{itemize}
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\item map Gaussian to Boltzmann
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\end{itemize}
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\end{frame}
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%================================================
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\section{Likelihood vs Score Matching}
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%================================================
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\begin{frame}{Flows}
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\[
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\max \log p(x)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Diffusion}
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\[
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\min \|s_\theta - \nabla \log p\|^2
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\]
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\end{frame}
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%================================================
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\section{Advantages and Limitations}
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%================================================
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\begin{frame}{Normalizing Flows}
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\begin{itemize}
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\item exact likelihood
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\item invertible
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\item limited flexibility
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Diffusion Models}
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\begin{itemize}
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\item highly expressive
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\item expensive sampling
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\item no explicit likelihood
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\end{itemize}
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\end{frame}
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%================================================
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\section{Geometric View}
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%================================================
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\begin{frame}{Flows}
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\begin{itemize}
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\item diffeomorphisms
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\item volume changes via Jacobian
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Diffusion}
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\begin{itemize}
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\item stochastic paths
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\item measure evolution
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\end{itemize}
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\end{frame}
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%================================================
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\section{Unification}
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%================================================
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\begin{frame}{Unified View}
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\[
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\text{Generative model} = \text{transport of probability measure}
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Operator Perspective}
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\begin{itemize}
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\item flows: pushforward operator
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\item diffusion: stochastic evolution operator
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Master Equation}
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\[
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\boxed{
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\text{Flows = deterministic transport}
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\quad
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\text{Diffusion = stochastic transport}
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}
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\]
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\end{frame}
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%================================================
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\section{Outlook}
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%================================================
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\begin{frame}{Future Directions}
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\begin{itemize}
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\item flow–diffusion hybrids
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\item Schrödinger bridges
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\item connections to quantum systems
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\end{itemize}
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\end{frame}
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\end{document}

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