scholarly journals High-performance sampling of generic determinantal point processes

2020 ◽  
Vol 378 (2166) ◽  
pp. 20190059 ◽  
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’.

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

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.

scholarly journals A note on the simulation of the Ginibre point process

2015 ◽  
Vol 52 (4) ◽  
pp. 1003-1012 ◽  
Author(s):  
Laurent Decreusefond ◽  
Ian Flint ◽  
Anais Vergne
Keyword(s):  
Point Process ◽  
Random Matrix ◽  
Point Processes ◽  
Matrix Theory ◽  
Applied Mathematics ◽  
Fixed Number ◽  
Extended Version ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
Ginibre Point Process

The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.

scholarly journals Reach of repulsion for determinantal point processes in high dimensions

2018 ◽  
Vol 55 (3) ◽  
pp. 760-788
Author(s):  
François Baccelli ◽  
Eliza O'Reilly
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Space Dimension ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
High Dimensions ◽  
Boolean Models ◽  
Moment Measure ◽  
First Moment ◽  
Typical Point

Abstract Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ0,! such that there exists a point process η where Φ=Φ0,!∪η in distribution and Φ0,!∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n→∞. We also prove that for some DPPs, there exists an R∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √nR∗. This R∗ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high-dimensional Boolean models is given.

scholarly journals Multi-Document Summarization with Determinantal Point Process Attention

2021 ◽  
Vol 71 ◽  
pp. 371-399
Author(s):  
Laura Perez-Beltrachini ◽  
Mirella Lapata
Keyword(s):  
Point Process ◽  
Experimental Evaluation ◽  
Point Processes ◽  
State Of The Art ◽  
The State ◽  
Attention Mechanism ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
Document Summarization

The ability to convey relevant and diverse information is critical in multi-document summarization and yet remains elusive for neural seq-to-seq models whose outputs are often redundant and fail to correctly cover important details. In this work, we propose an attention mechanism which encourages greater focus on relevance and diversity. Attention weights are computed based on (proportional) probabilities given by Determinantal Point Processes (DPPs) defined on the set of content units to be summarized. DPPs have been successfully used in extractive summarisation, here we use them to select relevant and diverse content for neural abstractive summarisation. We integrate DPP-based attention with various seq-to-seq architectures ranging from CNNs to LSTMs, and Transformers. Experimental evaluation shows that our attention mechanism consistently improves summarization and delivers performance comparable with the state-of-the-art on the MultiNews dataset

scholarly journals A note on the simulation of the Ginibre point process

2015 ◽  
Vol 52 (04) ◽  
pp. 1003-1012 ◽  
Author(s):  
Laurent Decreusefond ◽  
Ian Flint ◽  
Anais Vergne
Keyword(s):  
Point Process ◽  
Random Matrix ◽  
Point Processes ◽  
Matrix Theory ◽  
Applied Mathematics ◽  
Fixed Number ◽  
Extended Version ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
Ginibre Point Process

The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.

Examples of Stationary Sequences with Approximate Negative Dependence

2019 ◽  
pp. 305-318
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Spectral Density ◽  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Stationary Processes ◽  
Negative Dependence ◽  
Stationary Sequences ◽  
Determinantal Point Process ◽  
Dependent Random Variables ◽  
Associated Sequences

In this chapter, we treat several examples of stationary processes which are asymptotically negatively dependent and for which the results of Chapter 9 apply. Many systems in nature are complex, consisting of the contributions of several independent components. Our first examples are functions of two independent sequences, one negatively dependent and one interlaced mixing. For instance, the class of asymptotic negatively dependent random variables is used to treat functions of a determinantal point process and a Gaussian process with a positive continuous spectral density. Another example is point processes based on asymptotically negatively or positively associated sequences and displaced according to a Gaussian sequence with a positive continuous spectral density. Other examples include exchangeable processes, the weighted empirical process, and the exchangeable determinantal point process.

scholarly journals Random point processes and tilings arising from Gaussian analytic functions

2014 ◽  
Vol 70 (a1) ◽  
pp. C523-C523
Author(s):  
Michael Baake ◽  
Holger Koesters ◽  
Robert Moody
Keyword(s):  
Point Process ◽  
Analytic Functions ◽  
Point Processes ◽  
Logarithmic Potential ◽  
Basins Of Attraction ◽  
Random Point ◽  
Specific Class ◽  
Determinantal Point Processes ◽  
Point Sets ◽  
Gaussian Analytic Functions

Getting a grasp of what aperiodic order really entails is going to require collecting and understanding many diverse examples. Aperiodic crystals are at the top of the largely unknown iceberg beneath. Here we present a recently studied form of random point process in the (complex) plane which arises as the sets of zeros of a specific class of analytic functions given by power series with randomly chosen coefficients: Gaussian analytic functions (GAF). These point sets differ from Poisson processes by having a sort of built in repulsion between points, though the resulting sets almost surely fail both conditions of the Delone property. Remarkably the point sets that arise as the zeros of GAFs determine a random point process which is, in distribution, invariant under rotation and translation. In addition, there is a logarithmic potential function for which the zeros are the attractors, and the resulting basins of attraction produce tilings of the plane by tiles which are, almost surely, all of the same area. We discuss GAFs along with their tilings and diffraction, and as well note briefly their relationship to determinantal point processes, which are also of physical interest.

scholarly journals A Cellular Network Model with Ginibre Configured Base Stations

2014 ◽  
Vol 46 (3) ◽  
pp. 832-845 ◽  
Author(s):  
Naoto Miyoshi ◽  
Tomoyuki Shirai
Keyword(s):  
Point Process ◽  
Coverage Probability ◽  
Stochastic Geometry ◽  
Point Processes ◽  
Spatial Configuration ◽  
Base Stations ◽  
Determinantal Point Processes ◽  
Geometry Model ◽  
Wireless Nodes ◽  
Ginibre Point Process

Stochastic geometry models for wireless communication networks have recently attracted much attention. This is because the performance of such networks critically depends on the spatial configuration of wireless nodes and the irregularity of the node configuration in a real network can be captured by a spatial point process. However, most analysis of such stochastic geometry models for wireless networks assumes, owing to its tractability, that the wireless nodes are deployed according to homogeneous Poisson point processes. This means that the wireless nodes are located independently of each other and their spatial correlation is ignored. In this work we propose a stochastic geometry model of cellular networks such that the wireless base stations are deployed according to the Ginibre point process. The Ginibre point process is one of the determinantal point processes and accounts for the repulsion between the base stations. For the proposed model, we derive a computable representation for the coverage probability—the probability that the signal-to-interference-plus-noise ratio (SINR) for a mobile user achieves a target threshold. To capture its qualitative property, we further investigate the asymptotics of the coverage probability as the SINR threshold becomes large in a special case. We also present the results of some numerical experiments.

scholarly journals Exact sampling of determinantal point processes without eigendecomposition

2020 ◽  
Vol 57 (4) ◽  
pp. 1198-1221
Author(s):  
Claire Launay ◽  
Bruno Galerne ◽  
Agnès Desolneux
Keyword(s):  
Spectral Decomposition ◽  
Numerical Experiments ◽  
Point Processes ◽  
Exact Algorithm ◽  
Determinantal Point Processes ◽  
Exact Sampling ◽  
Discrete Spaces ◽  
Discrete Setting

AbstractDeterminantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel K that can be seen, in a discrete setting, as a matrix storing the similarity between points. The main exact algorithm to sample DPPs uses the spectral decomposition of K, a computation that becomes costly when dealing with a high number of points. Here we present an alternative exact algorithm to sample in discrete spaces that avoids the eigenvalues and the eigenvectors computation. The method used here is innovative, and numerical experiments show competitive results with respect to the initial algorithm.

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