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Towards a Compositional Framework for Convex Analysis (with Applications to Probability Theory)

Dario Stein, Richard Samuelson

Abstract

We introduce a compositional framework for convex analysis based on the notion of convex bifunction of Rockafellar. This framework is well-suited to graphical reasoning, and exhibits rich dualities such as the Legendre-Fenchel transform, while generalizing formalisms like graphical linear algebra, convex relations and convex programming. We connect our framework to probability theory by interpreting the Laplace approximation in its context: The exactness of this approximation on normal distributions means that logdensity is a functor from Gaussian probability (densities and integration) to concave bifunctions and maximization.

Towards a Compositional Framework for Convex Analysis (with Applications to Probability Theory)

Abstract

We introduce a compositional framework for convex analysis based on the notion of convex bifunction of Rockafellar. This framework is well-suited to graphical reasoning, and exhibits rich dualities such as the Legendre-Fenchel transform, while generalizing formalisms like graphical linear algebra, convex relations and convex programming. We connect our framework to probability theory by interpreting the Laplace approximation in its context: The exactness of this approximation on normal distributions means that logdensity is a functor from Gaussian probability (densities and integration) to concave bifunctions and maximization.
Paper Structure (5 sections, 4 equations, 1 figure)

This paper contains 5 sections, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Categorical Methods
  3. Applications to Probability Theory
  4. Contribution and Outline
  5. Overview of Convex Analysis

Figures (1)

  • Figure 1: Addition of independent normal variables $X,Y$. Left: pdf and convolution, right: logpdf and sup-convolution