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Dimension-Free Descriptions of Convex Sets

Eitan Levin, Venkat Chandrasekaran

TL;DR

The paper develops a systematic framework for dimension-free descriptions of convex sets and related functions, grounded in representation stability. By organizing sets as consistent sequences with freely-described elements, it derives conditions for compatibility across dimensions, extendability to freely-described sequences, and limiting continuous descriptions. It shows that many invariant optimization problems admit constant-sized reductions in high dimension, unifying results across extremal combinatorics, quantum information, and graph theory, and it provides a data-driven method to learn dimension-free convex descriptions (including convex regression) that extend across sizes. The approach yields practical semidefinite representations and parametric families tailored to applications, with detailed examples (simplices, permutahedra, free spectrahedra, graphon parameters) and a computational pipeline for learning and extending across dimensions. This work thus forges a bridge between representation stability and convex geometry, enabling scalable, dimension-agnostic optimization and learning across size-parameterized problem families.

Abstract

Convex sets arising in a variety of applications are well-defined for every relevant dimension. Examples include the simplex and the spectraplex that correspond to probability distributions and to quantum states; combinatorial polytopes and their relaxations such as the cut polytope and the elliptope in integer programming; and unit balls of regularizers such as the $\ell_p$ and Schatten norms in inverse problems. Moreover, these sets are often specified using conic descriptions that can be obviously instantiated in any dimension. We develop a systematic framework to study such dimension-free descriptions of convex sets. We show that dimension-free descriptions arise from a recently-identified phenomenon in algebraic topology called representation stability, which relates invariants across dimensions in a sequence of group representations. Our framework yields structural results for dimension-free descriptions pertaining to the relations between the sets they describe across dimensions, extendability of a single set in a given dimension to a freely-described sequence, and continuous limits of such sequences. We also develop a procedure to obtain parametric families of freely-described convex sets whose structure is adapted to a given application; illustrations are provided via examples that arise in the literature as well as new families that are derived using our procedure. We demonstrate the utility of our framework in two contexts. First, we develop an algorithm for a dimension-free analog of the convex regression problem, where a convex function is fit to input-output data; by searching over our parametric families, we can fit a function to low-dimensional inputs and extend it to any other dimension. Second, we prove that many sequences of symmetric conic programs can be solved in constant time, which unifies and strengthens several results in the literature.

Dimension-Free Descriptions of Convex Sets

TL;DR

The paper develops a systematic framework for dimension-free descriptions of convex sets and related functions, grounded in representation stability. By organizing sets as consistent sequences with freely-described elements, it derives conditions for compatibility across dimensions, extendability to freely-described sequences, and limiting continuous descriptions. It shows that many invariant optimization problems admit constant-sized reductions in high dimension, unifying results across extremal combinatorics, quantum information, and graph theory, and it provides a data-driven method to learn dimension-free convex descriptions (including convex regression) that extend across sizes. The approach yields practical semidefinite representations and parametric families tailored to applications, with detailed examples (simplices, permutahedra, free spectrahedra, graphon parameters) and a computational pipeline for learning and extending across dimensions. This work thus forges a bridge between representation stability and convex geometry, enabling scalable, dimension-agnostic optimization and learning across size-parameterized problem families.

Abstract

Convex sets arising in a variety of applications are well-defined for every relevant dimension. Examples include the simplex and the spectraplex that correspond to probability distributions and to quantum states; combinatorial polytopes and their relaxations such as the cut polytope and the elliptope in integer programming; and unit balls of regularizers such as the and Schatten norms in inverse problems. Moreover, these sets are often specified using conic descriptions that can be obviously instantiated in any dimension. We develop a systematic framework to study such dimension-free descriptions of convex sets. We show that dimension-free descriptions arise from a recently-identified phenomenon in algebraic topology called representation stability, which relates invariants across dimensions in a sequence of group representations. Our framework yields structural results for dimension-free descriptions pertaining to the relations between the sets they describe across dimensions, extendability of a single set in a given dimension to a freely-described sequence, and continuous limits of such sequences. We also develop a procedure to obtain parametric families of freely-described convex sets whose structure is adapted to a given application; illustrations are provided via examples that arise in the literature as well as new families that are derived using our procedure. We demonstrate the utility of our framework in two contexts. First, we develop an algorithm for a dimension-free analog of the convex regression problem, where a convex function is fit to input-output data; by searching over our parametric families, we can fit a function to low-dimensional inputs and extend it to any other dimension. Second, we prove that many sequences of symmetric conic programs can be solved in constant time, which unifies and strengthens several results in the literature.
Paper Structure (60 sections, 41 theorems, 79 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 60 sections, 41 theorems, 79 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Framework and Contributions
  3. Freely-described Convex Sets
  4. Structural Aspects of dimension-free descriptions
  5. Invariant Conic Programs
  6. Dimension-Free Convex Regression
  7. Related Work
  8. Extended formulations of convex sets:
  9. Free spectrahedra, noncommutative convex algebraic geometry:
  10. Representation stability:
  11. Notation and Basics
  12. Basics:
  13. Representation theory:
  14. Convex sets and functions:
  15. Background on Representation Stability
  16. ...and 45 more sections

Key Result

Theorem 1.11

Consider the consistent sequence $\mathscr{V}_0=\{\mathbb{R}^n\}$ with embeddings by zero-padding and the standard actions of one of the sequences of classical Weyl groups as in Example ex:basic_consist_seq. For any consistent sequence $\mathscr{V}=\{\mathbb{V}_n\}$ obtained from $\mathscr{V}_0$ by

Figures (2)

  • Figure 1: Errors for learning the $\ell_{\pi}$ norm and the quantum entropy variant \ref{['eq:quant_ent_mod']}. The dashed vertical lines denote the max $n$ for which data is available.
  • Figure 2: Weighted undirected graph represented as a graph, an adjacency matrix $X$, and a symmetric function (graphon) $W_X$ on $[0,1]^2$.

Theorems & Definitions (108)

  • Example 1.1
  • Example 1.2
  • Definition 1.3: Consistent sequences CHURCH2013250
  • Definition 1.4: Freely-described elements
  • Example 1.5: Vectors with zero-padding
  • Definition 1.6: Dimension-free conic descriptions
  • Example 1.7: Parametric convex graph invariants
  • Definition 1.8: Compatibility conditions
  • Example 1.9: Convex graphon parameters
  • Definition 1.10: Constant-sized descriptions
  • ...and 98 more