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Determinantal Point Processes in the Flat Limit: Extended L-ensembles, Partial-Projection DPPs and Universality Classes by Barthelmé, Simon; Tremblay, Nicolas; Usevich, Konstantin; Amblard, Pierre-Olivier is a scholarly article available to read on EtoBox.

What is Determinantal Point Processes in the Flat Limit: Extended L-ensembles, Partial-Projection DPPs and Universality Classes about?

Determinantal point processes (DPPs) are repulsive point processes where the interaction between points depends on the determinant of a positive-semi definite matrix. The contributions of this paper are two-fold. First of all, we introduce the concept of extended L-ensemble, a novel representation of DPPs. These extended L-ensembles are interesting objects because they fix some pathologies in the usual formalism of DPPs, for instance the fact that projection DPPs are not L-ensembles. Every (fixed-size) DPP is an (fixed-size) extended L-ensemble, including projection DPPs. This new formalism enables to introduce and analyze a subclass of DPPs, called partial-projection DPPs. Secondly, with these new definitions in hand, we first show that partial-projection DPPs arise as perturbative limits of L-ensembles, that is, limits in $\varepsilon \rightarrow 0$ of L-ensembles based on matrices of the form $\varepsilon \mathbf{A} + \mathbf{B}$ where $\mathbf{B}$ is low-rank. We generalise this result by showing that partial-projection DPPs also arise as the limiting process of L-ensembles based on kernel matrices, when the kernel function becomes flat (so that every point interacts with every

Author

Barthelmé, Simon; Tremblay, Nicolas; Usevich, Konstantin; Amblard, Pierre-Olivier

Published

2020

Language

EN