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‎doc/src/week12/week12.do.txt‎

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@@ -7,7 +7,7 @@ DATE: April 16, 2026
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!bblock Generative methods, energy models and Boltzmann machines
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o Summary of discussions on Restricted Boltzmann machines, reminder from last week
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o Introduction to Variational Autoencoders (VAEs), basic mathematical formalims
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o Introduction to Variational Autoencoders (VAEs), basic mathematical formalism
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#o "Video of lecture":"https://youtu.be/Mm9Xasy8qNw"
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#o "Whiteboard notes":"https://github.com/CompPhysics/AdvancedMachineLearning/blob/main/doc/HandwrittenNotes/2025/NotesApril10.pdf"
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!eblock

‎doc/src/week15/Latexfiles/normalizing.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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\usepackage{graphicx}
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\usetheme{Madrid}
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\title{Normalizing Flows}
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\subtitle{Probability Transport and Invertible Models}
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\author{Morten Hjorth-Jensen}
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\date{Spring 2026}
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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 Problem}
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We wish to model a probability density:
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\[
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p(x), \quad x \in \mathbb{R}^d
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\]
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Goal:
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\begin{itemize}
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\item tractable likelihood
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\item efficient sampling
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\item expressive model
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Key Idea}
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Start with simple distribution:
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\[
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z \sim p_0(z), \quad p_0 = \mathcal{N}(0,I)
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\]
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Apply invertible map:
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\[
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x = f(z)
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\]
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Then:
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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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\section{Change of Variables}
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%================================================
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\begin{frame}{Change of Variables Theorem}
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For invertible $f$:
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\[
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x = f(z)
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\]
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\[
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p(x) = p_0(z)\left|\det \frac{\partial f^{-1}}{\partial x}\right|
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\]
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Equivalently:
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\[
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p(x) = p_0(z)\left|\det \frac{\partial f}{\partial z}\right|^{-1}
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Log-Density}
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\[
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\log p(x) = \log p_0(z) - \log \left| \det \frac{\partial f}{\partial z} \right|
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\]
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\end{frame}
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%================================================
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\section{Composition of Flows}
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%================================================
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\begin{frame}{Flow Model}
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Define:
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\[
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x = f_K \circ f_{K-1} \circ \cdots \circ f_1(z)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Log-Likelihood}
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\[
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\log p(x) = \log p_0(z) - \sum_{k=1}^K \log \left|\det \frac{\partial f_k}{\partial h_{k-1}}\right|
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\]
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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 Each layer transports probability mass
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\item Total transformation = composition
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\end{itemize}
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\end{frame}
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%================================================
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\section{Simple Flow Layers}
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%================================================
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\begin{frame}{Affine Transformation}
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\[
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x = Az + b
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\]
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\[
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\det J = \det A
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Planar Flow}
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\[
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f(z) = z + u \tanh(w^T z + b)
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\]
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Jacobian:
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\[
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\det J = 1 + u^T \psi(z)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Radial Flow}
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\[
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f(z) = z + \beta \frac{z - z_0}{\alpha + \|z - z_0\|}
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\]
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\end{frame}
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%================================================
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\section{Coupling Layers}
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%================================================
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\begin{frame}{RealNVP Idea}
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Split variables:
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\[
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z = (z_1, z_2)
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\]
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Transform:
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\[
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x_1 = z_1, \quad x_2 = z_2 \odot e^{s(z_1)} + t(z_1)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Jacobian}
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\[
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\det J = \exp\left(\sum s(z_1)\right)
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\]
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\begin{itemize}
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\item triangular Jacobian → efficient
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\end{itemize}
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\end{frame}
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%================================================
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\section{Autoregressive Flows}
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%================================================
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\begin{frame}{Autoregressive Model}
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\[
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x_i = f_i(z_i; x_1,\dots,x_{i-1})
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Jacobian}
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\[
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\det J = \prod_i \frac{\partial x_i}{\partial z_i}
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\]
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\end{frame}
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%================================================
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\section{Continuous Normalizing Flows}
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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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\end{frame}
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%------------------------------------------------
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\begin{frame}{Density Evolution}
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\[
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\frac{d}{dt} \log p(x(t)) = -\nabla \cdot v(x,t)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Connection to Physics}
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\begin{itemize}
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\item continuity equation
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\item Liouville equation
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\end{itemize}
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\end{frame}
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%================================================
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\section{Geometric Interpretation}
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%================================================
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\begin{frame}{Flows as Diffeomorphisms}
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\[
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f: \mathbb{R}^d \rightarrow \mathbb{R}^d
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\]
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\begin{itemize}
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\item invertible
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\item smooth
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Manifold View}
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\begin{itemize}
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\item probability distributions as densities on manifold
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\item flow = transport map
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\end{itemize}
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\end{frame}
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%================================================
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\section{Connection to Statistical Physics}
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%================================================
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\begin{frame}{Boltzmann Distribution}
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\[
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p(x) = \frac{1}{Z} e^{-E(x)}
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Partition Function Problem}
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\[
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Z = \int e^{-E(x)} dx
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\]
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\begin{itemize}
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\item hard to compute
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Flow Interpretation}
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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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\begin{itemize}
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\item replaces partition function with Jacobian
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\end{itemize}
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\end{frame}
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%================================================
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\section{Optimal Transport Connection}
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%================================================
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\begin{frame}{Transport Map}
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\[
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T: p_0(z) \rightarrow p(x)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Monge Problem}
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\[
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\min_T \int \|x - T(x)\|^2 p_0(x) dx
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Relation to Flows}
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\begin{itemize}
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\item flows approximate transport maps
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\item connect to Wasserstein geometry
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\end{itemize}
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\end{frame}
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%================================================
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\section{Training}
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%================================================
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\begin{frame}{Maximum Likelihood}
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\[
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\mathcal{L} = \sum_i \log p(x_i)
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Gradient}
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\[
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\nabla_\theta \log p(x) =
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\nabla_\theta \log p_0(z) -
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\nabla_\theta \log |\det J|
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\]
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\end{frame}
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%================================================
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\section{Limitations}
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%================================================
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\begin{frame}{Challenges}
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\begin{itemize}
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\item invertibility constraint
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\item Jacobian computation
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\item expressivity vs tractability
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\end{itemize}
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\end{frame}
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%================================================
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\section{Summary}
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%================================================
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\begin{frame}{Summary}
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\begin{itemize}
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\item normalizing flows = invertible transformations
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\item exact likelihood computation
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\item deep connection to physics and geometry
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\end{itemize}
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\end{frame}
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\end{document}

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