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Application and issues in abstract convexity

Reinier Díaz Millán, Nadezda Sukhorukova, Julien Ugon

TL;DR

The paper investigates using abstract convexity, i.e., convexity without linearity, to perform uniform Chebyshev approximation by replacing linear elementary components with $L$-convex (or quasiaffine) objects. It highlights a key result: the set of lower semicontinuous quasiaffine functions forms a supremal generator for lower semicontinuous quasiconvex functions, ensuring sublevel sets are half-spaces and enabling tractable optimization. A bisection framework for quasiconvex approximation is developed, where each step requires checking the non-emptiness of a polytope, reducible to linear programs when the approximation domain $X$ is a finite grid. Numerical experiments demonstrate practical approximations using compositions of monotone functions with affine or rational forms, with implications for deep learning and connections to axiomatic convexity, suggesting broad applicability beyond theory.

Abstract

The theory of abstract convexity, also known as convexity without linearity, is an extension of the classical convex analysis. There are a number of remarkable results, mostly concerning duality, and some numerical methods, however, this area has not found many practical applications yet. In this paper we study the application of abstract convexity to function approximation. Another important research direction addressed in this paper is the connection with the so-called axiomatic convexity.

Application and issues in abstract convexity

TL;DR

The paper investigates using abstract convexity, i.e., convexity without linearity, to perform uniform Chebyshev approximation by replacing linear elementary components with -convex (or quasiaffine) objects. It highlights a key result: the set of lower semicontinuous quasiaffine functions forms a supremal generator for lower semicontinuous quasiconvex functions, ensuring sublevel sets are half-spaces and enabling tractable optimization. A bisection framework for quasiconvex approximation is developed, where each step requires checking the non-emptiness of a polytope, reducible to linear programs when the approximation domain is a finite grid. Numerical experiments demonstrate practical approximations using compositions of monotone functions with affine or rational forms, with implications for deep learning and connections to axiomatic convexity, suggesting broad applicability beyond theory.

Abstract

The theory of abstract convexity, also known as convexity without linearity, is an extension of the classical convex analysis. There are a number of remarkable results, mostly concerning duality, and some numerical methods, however, this area has not found many practical applications yet. In this paper we study the application of abstract convexity to function approximation. Another important research direction addressed in this paper is the connection with the so-called axiomatic convexity.
Paper Structure (15 sections, 6 theorems, 32 equations, 5 figures)

This paper contains 15 sections, 6 theorems, 32 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Abstract Convexity
  3. Definitions and preliminaries
  4. Quasiconvexity
  5. Quasiconvex functions and supremal generators
  6. Relation to Axiomatic Convexity
  7. Application to approximation
  8. Problem formulation
  9. Bisection method
  10. Approximation by a quasiaffine function
  11. Numerical experiments
  12. Strictly monotone function in composition with an affine function.
  13. Strictly monotone function in composition with a rational function.
  14. Deep learning applications.
  15. Conclusions

Key Result

Theorem 2.8

The set of all lower semicontinuous quasiaffine functions forms a supremal generator of the set of all lower semicontinuous quasiconvex functions.

Figures (5)

  • Figure 1: Function $f(x,y)=(-x+y^3+x^4)^4$.
  • Figure 2: Approximation $g(A,x,y)=(a_1+a_2x+a_3y+a_4x^2+a_5y^2+a_6xy)^3$.
  • Figure 3: Deviation (error) function $f(x,y)-g(A,x,y)=(-x+y^3+x^4)^4-(a_1+a_2x+a_3y+a_4x^2+a_5y^2+a_6xy)^3.$
  • Figure 4: Approximation $g_1(A,x,y)=(\frac{a_1+a_2x+a_3y+a_4x^2+a_5y^2}{1+a_6xy})^3$.
  • Figure 5: Deviation (error) function $f(x,y)-g(A,x,y)=(-x+y^3+x^4)^4-\left(\frac{a_1+a_2x+a_3y+a_4x^2+a_5y^2}{1+a_6xy}\right)^3.$

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2: Abstract Convexity Rubinov00
  • Definition 2.3: Support Set Rubinov00
  • Definition 2.4: Supremal generator dutta2005abstractRubinov00
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Proposition 2.9
  • proof
  • ...and 12 more