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.
Highlights
A large class of problems in random geometry is concerned with the collocation of points in high-dimensional space
We have reviewed various problems from different disciplines, including high-dimensional random geometry, finance, binary classification with a perceptron, game theory, and random linear algebra, which all have at their root the problem of dichotomies, that is, the linear separability of points carrying a binary label and scattered randomly over a high-dimensional space
No doubt there are several further problems belonging to this class; those that spring to mind are theoretical ecology alluded to at the end of the previous Section, or linear programming with random parameters [8]
Summary
A large class of problems in random geometry is concerned with the collocation of points in high-dimensional space. It is frequently relevant to consider the case where both the number of points T and the dimension of space N tend to infinity. This limit is often characterized by abrupt qualitative changes reminiscent of phase transitions when an external parameter or the ratio T/N vary and cross a critical value. We point out the connection with the existence of non-negative solutions to systems of linear equations and with mixed strategies in zero-sum games. In addition to uncovering the common random geometrical background of seemingly very different problems, our comparative analysis sheds light on each of them from various angles and points to ramifications in their respective fields
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