Dimension-Free Descriptions of Convex Sets
Eitan Levin, Venkat Chandrasekaran

TL;DR
This paper introduces a framework based on representation stability for understanding and constructing dimension-free descriptions of convex sets, enabling scalable algorithms and unified solutions across dimensions.
Contribution
It develops a systematic approach to dimension-free convex set descriptions, linking algebraic topology with convex geometry, and provides practical algorithms and unified solution methods.
Findings
Dimension-free descriptions relate to representation stability in algebraic topology.
The framework allows construction of parametric families of convex sets for various applications.
Many symmetric conic programs can be solved in constant time using the proposed approach.
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…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Advanced Optimization Algorithms Research · Point processes and geometric inequalities
