Aspects of a phase transition in high-dimensional random geometry

TL;DR

This paper investigates a phase transition phenomenon in high-dimensional random geometry, revealing its implications across diverse fields such as finance, machine learning, and ecology, and establishing connections between seemingly unrelated problems.

Contribution

It provides a unified analysis of the geometric phase transition in high-dimensional spaces and links various applications across different scientific domains.

Findings

Identifies conditions for phase transition in high-dimensional problems

Connects geometric phase transitions to practical problems in finance and ecology

Offers a framework for analyzing solvability in high-dimensional systems

Abstract

A phase transition in high-dimensional random geometry is analyzed as it arises in a variety of problems. A prominent example is the feasibility of a minimax problem that represents the extremal case of a class of financial risk measures, among them the current regulatory market risk measure Expected Shortfall. Others include portfolio optimization with a ban on short selling, the storage capacity of the perceptron, the solvability of a set of linear equations with random coefficients, and competition for resources in an ecological system. These examples shed light on various aspects of the underlying geometric phase transition, create links between problems belonging to seemingly distant fields and offer the possibility for further ramifications.