Determinantal point processes in the plane from products of random matrices

Abstract Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials, we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part, we introduce invertible polynomial ensembles given, e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by Fyodorov, Grela, and Strahov. for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble.

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Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.123 ◽

2020 ◽

pp. 1-14

Author(s):

SHOTA OSADA

Keyword(s):

Point Processes ◽

Duke Math ◽

Random Point ◽

Determinantal Point Processes ◽

Fredholm Determinants ◽

Poisson Point Processes ◽

Translation Invariant ◽

Ergodic Properties ◽

Tree Representations ◽

Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

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Injecting Semantic Diversity in Top-N Recommender Systems Using Determinantal Point Processes and Curated Lists

Adjunct Publication of the 26th Conference on User Modeling, Adaptation and Personalization - UMAP '18 ◽

10.1145/3213586.3226223 ◽

2018 ◽

Author(s):

Surya Kallumadi ◽

Gabriel Necoechea

Keyword(s):

Recommender Systems ◽

Point Processes ◽

Determinantal Point Processes

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DPP-VSE: Constructing a variable selection ensemble by determinantal point processes

Expert Systems with Applications ◽

10.1016/j.eswa.2021.115025 ◽

2021 ◽

Vol 178 ◽

pp. 115025

Author(s):

Chunxia Zhang ◽

Junmin Liu ◽

Guanwei Wang ◽

Guanghai Li

Keyword(s):

Variable Selection ◽

Point Processes ◽

Determinantal Point Processes ◽

Variable Selection Ensemble

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Asymptotic behaviour of products of random matrices

Probability Theory and Applications ◽

10.1515/9783112314227-101 ◽

1987 ◽

pp. 743-748

Keyword(s):

Asymptotic Behaviour ◽

Random Matrices ◽

Products Of Random Matrices

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Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

Journal of Applied Probability ◽

10.1017/jpr.2020.101 ◽

2021 ◽

Vol 58 (2) ◽

pp. 469-483

Author(s):

Jesper Møller ◽

Eliza O’Reilly

Keyword(s):

Finite Number ◽

Point Process ◽

Point Processes ◽

Parametric Models ◽

Determinantal Point Processes ◽

Determinantal Point Process ◽

The Difference ◽

Weaker Conditions ◽

Palm Distributions

AbstractFor a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)>0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

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High-performance sampling of generic determinantal point processes

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2019.0059 ◽

2020 ◽

Vol 378 (2166) ◽

pp. 20190059 ◽

Cited By ~ 1

Author(s):

Jack Poulson

Keyword(s):

Point Process ◽

Spectral Decomposition ◽

High Performance ◽

Point Processes ◽

Cholesky Factorization ◽

Determinantal Point Processes ◽

Determinantal Point Process ◽

Decomposition Approach ◽

Map Inference ◽

Sampling Schemes

Determinantal point processes (DPPs) were introduced by Macchi (Macchi 1975 Adv. Appl. Probab. 7 , 83–122) as a model for repulsive (fermionic) particle distributions. But their recent popularization is largely due to their usefulness for encouraging diversity in the final stage of a recommender system (Kulesza & Taskar 2012 Found. Trends Mach. Learn. 5 , 123–286). The standard sampling scheme for finite DPPs is a spectral decomposition followed by an equivalent of a randomly diagonally pivoted Cholesky factorization of an orthogonal projection, which is only applicable to Hermitian kernels and has an expensive set-up cost. Researchers Launay et al. 2018 ( http://arxiv.org/abs/1802.08429 ); Chen & Zhang 2018 NeurIPS ( https://papers.nips.cc/paper/7805-fast-greedy-map-inference-for-determinantal-point-process-to-improve-recommendation-diversity.pdf ) have begun to connect DPP sampling to LDL H factorizations as a means of avoiding the initial spectral decomposition, but existing approaches have only outperformed the spectral decomposition approach in special circumstances, where the number of kept modes is a small percentage of the ground set size. This article proves that trivial modifications of LU and LDL H factorizations yield efficient direct sampling schemes for non-Hermitian and Hermitian DPP kernels, respectively. Furthermore, it is experimentally shown that even dynamically scheduled, shared-memory parallelizations of high-performance dense and sparse-direct factorizations can be trivially modified to yield DPP sampling schemes with essentially identical performance. The software developed as part of this research, Catamari ( hodgestar.com/catamari ) is released under the Mozilla Public License v.2.0. It contains header-only, C++14 plus OpenMP 4.0 implementations of dense and sparse-direct, Hermitian and non-Hermitian DPP samplers. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

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