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

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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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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