Determinantal Point Processes Research Articles

Entanglement entropy and hyperuniformity of Ginibre and Weyl–Heisenberg ensembles

We show that, for a class of planar determinantal point processes (DPP) {mathcal {X}}, the growth of the entanglement entropy S({mathcal {X}}(Omega )) of {mathcal {X}} on a compact region Omega subset {mathbb {R}}^{2d}, is related to the variance {mathbb {V}}left( {mathcal {X}}(Omega )right) as follows: VX(Ω)≲SX(Ω)≲VX(Ω).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathbb {V}}\\left( {\\mathcal {X}}(\\Omega )\\right) \\lesssim S\\left( \\mathcal {X(} \\Omega \\mathcal {)}\\right) \\lesssim {\\mathbb {V}}\\left( {\\mathcal {X}}(\\Omega )\\right) . \\end{aligned}$$\\end{document}Therefore, such DPPs satisfy an area lawSleft( {mathcal {X}}_{g} mathcal {(}Omega mathcal {)}right) lesssim left| partial {Omega } right| , where partial {Omega } is the boundary of Omega if they are of Class I hyperuniformity ({mathbb {V}}left( {mathcal {X}} (Omega )right) lesssim left| partial {Omega }right| ), while the area law is violated if they are of Class II hyperuniformity (as Lrightarrow infty , {mathbb {V}}left( {mathcal {X}} (LOmega )right) sim C_{Omega }L^{d-1}log L). As a result, the entanglement entropy of Weyl–Heisenberg ensembles (a family of DPPs containing the Ginibre ensemble and Ginibre-type ensembles in higher Landau levels), satisfies an area law, as a consequence of its hyperuniformity.

Determinantal point processes in the flat limit

Determinantal point processes (DPPs) are repulsive point processes where the interaction between points depends on the determinant of a positive-semi definite matrix. In this paper, we study the limiting process of L-ensembles based on kernel matrices, when the kernel function becomes flat (so that every point interacts with every other point, in a sense). We show that these limiting processes are best described in the formalism of extended L-ensembles and partial projection DPPs, and the exact limit depends mostly on the smoothness of the kernel function. In some cases, the limiting process is even universal, meaning that it does not depend on specifics of the kernel function, but only on its degree of smoothness. Since flat-limit DPPs are still repulsive processes, this implies that practically useful families of DPPs exist that do not require a spatial length-scale parameter.

The spherical ensemble and quasi-Monte-Carlo designs

The spherical ensemble is a well-known ensemble of N repulsive points on the two-dimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall state...). Here we show that the spherical ensemble enjoys remarkable convergence properties from the point of view of numerical integration. More precisely, it is shown that the numerical integration rule corresponding to N nodes on the two-dimensional sphere sampled in the spherical ensemble is, with overwhelming probability, nearly a quasi-Monte-Carlo design in the sense of Brauchart-Saff-Sloan-Womersley for any smoothness parameter s≤2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\le 2.$$\\end{document} The key ingredient is a new explicit sub-Gaussian concentration of measure inequality for the spherical ensemble.

The affine ensemble: determinantal point processes associated with the $ax + b$ group

We introduce the affine ensemble, a class of determinantal point processes (DPP) in the half-plane $\mathbb{C}^{+}$ associated with the $ax + b$ (affine) group, depending on an admissible Hardy function $\psi$. We obtain the asymptotic behavior of the variance, the exact value of the asymptotic constant, and non-asymptotic upper and lower bounds for the variance on a compact set $\Omega \subset \mathbb{C}^{+}$. As a special case one recovers the DPP related to the weighted Bergman kernel. When $\psi$ is chosen within a finite family whose Fourier transform are Laguerre functions, we obtain the DPP associated to hyperbolic Landau levels, the eigenspaces of the finite spectrum of the Maass Laplacian with a magnetic field.

Interference Analysis in Non-Poisson Networks Under Spatially Correlated Shadowing

Spatial correlations that appear in mobile wireless networks, such as the correlations of node locations and shadowing effects, significantly affect the characteristics of interference, which may degrade the performance of various wireless network systems. In this paper, we theoretically analyze the statistical and temporal characteristics of interference in a network where the node locations and shadowing are spatially correlated. We model the correlation of the node locations by two types of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">non-Poisson</i> point processes: determinantal point processes (DPPs) for modeling <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">repulsiveness</i> and doubly Poisson cluster processes (DPCPs) for modeling <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">attractiveness</i> . Furthermore, we consider Gudmundson's model for the spatial correlation of shadowing. Using this model and assuming an i.i.d. mobility of nodes, we analyze the variance along with the spatial and temporal correlations of interference. Since the exact expressions are in non-analytical forms, we derive their simple closed-form asymptotic formulas when the variance of the shadowing is large. The results show a readable relationship between the characteristics of interference and various system parameters. This relationship can be used for realistic modeling and better understanding of various wireless network systems under spatially correlated shadowing. Moreover, we discuss various numerical examples and demonstrate that the obtained asymptotic expressions achieve tight approximation.

Large gap asymptotics on annuli in the random normal matrix model

We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the form exp(C1n2+C2nlogn+C3n+C4n+C5logn+C6+Fn+O(n-112)),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\exp \\Bigg ( C_{1} n^{2} + C_{2} n \\log n + C_{3} n + C_{4} \\sqrt{n} + C_{5}\\log n + C_{6} + {\\mathcal {F}}_{n} + \\mathcal {O}\\Big ( n^{-\\frac{1}{12}}\\Big )\\Bigg ), \\end{aligned}$$\\end{document}where n is the number of points of the process. We determine the constants C1,…,C6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},\\ldots ,C_{6}$$\\end{document} explicitly, as well as the oscillatory term Fn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{n}$$\\end{document} which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only C1,…,C4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},\\ldots ,C_{4}$$\\end{document} were previously known, (ii) when the hole region is an unbounded annulus, only C1,C2,C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},C_{2},C_{3}$$\\end{document} were previously known, and (iii) when the hole region is a regular annulus in the bulk, only C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1}$$\\end{document} was previously known. For general values of our parameters, even C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1}$$\\end{document} is new. A main discovery of this work is that Fn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{n}$$\\end{document} is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.

Personalized diversification of complementary recommendations with user preference in online grocery.

Complementary recommendations play an important role in surfacing the relevant items to the customers. In the cross-selling scenario, some customers might present more exploratory shopping behaviors and prefer more diverse complements, while other customers show less exploratory (or more conventional) shopping behaviors and want to have a deep dive of less diverse types of complements. The existence of two distinct shopping behaviors reflects users' different shopping intents and requires complementary recommendations to be adaptable based on the user's shopping intent. Although many studies focus on improving the recommendations through post-processing techniques, such as user-item-level personalized ranking and diversification of recommendations, they fail to address such a requirement. First, many user-item-level personalization methods cannot explicitly model the preference of users in two types of shopping behaviors and their intent on the corresponding complementary recommendations. Second, most of the diversification methods increase the heterogeneity of the recommendations. However, users' intent on conventional complementary shopping requires more homogeneity of the recommendations, which is not explicitly modeled. The present study tries attempts to solve these problems by the personalized diversification strategies for complementary recommendations. To address the requirement of modeling heterogenized and homogenized complementary recommendations, we propose two diversification strategies, heterogenization and homogenization, to re-rank complementary recommendations based on the determinantal point process (DPP). We use transaction history to estimate users' intent on more exploratory or more conventional complementary shopping. With the estimated user intent scores and two diversification strategies, we propose an algorithm to personalize the diversification strategies dynamically. We demonstrate the effectiveness of our re-ranking algorithm on the publicly available Instacart dataset.

Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance property is developed. This involves isolating a class of invariant unitary matrices, to be referred to as cyclic Pólya ensembles, and examining their properties with respect to the spherical transform on $$\mathrm U(N)$$ . Included in the cyclic Pólya ensemble class are Haar invariant unitary matrices, the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, and give a uniqueness statement for the corresponding weights. The natural appearance of bilateral hypergeometric series is highlighted, with this special function playing the role of the Meijer G-function in the transform theory of unitary invariant product of positive definite matrices. We construct a family of functions forming bi-orthonormal pairs which underly the correlation kernel of the corresponding determinantal point processes, and furthermore obtain an integral formula for the correlation kernel involving just two of these functions.

Double interlacing in random tiling models

Random tilings of very large domains will typically lead to a solid, a liquid, and a gas phase. In the two-phase case, the solid–liquid boundary (arctic curve) is smooth, possibly with singularities. At the point of tangency of the arctic curve with the domain boundary, for large-sized domains, the tiles of a certain shape form a singly interlacing set, fluctuating according to the eigenvalues of the principal minors of a Gaussian unitary ensemble-matrix. Introducing non-convexities in large domains may lead to the appearance of several interacting liquid regions: They can merely touch, leading to either a split tacnode (hard tacnode), with two distinct adjacent frozen phases descending into the tacnode, or a soft tacnode. For appropriate scaling of the non-convex domains and probing about such split tacnodes, filaments, evolving in a bricklike sea of dimers of another type, will connect the liquid patches. Nearby, the tiling fluctuations are governed by a discrete tacnode kernel—i.e., a determinantal point process on a doubly interlacing set of dots belonging to a discrete array of parallel lines. This kernel enables us to compute the joint distribution of the dots along those lines. This kernel appears in two very different models: (i) domino tilings of skew-Aztec rectangles and (ii) lozenge tilings of hexagons with cuts along opposite edges. Soft tacnodes appear when two arctic curves gently touch each other amid a bricklike sea of dimers of one type, unlike the split tacnode. We hope that this largely expository paper will provide a view on the subject and be accessible to a wider audience.

Extended L-ensembles: A new representation for determinantal point processes

Determinantal point processes (DPPs) are a class of repulsive point processes, popular for their relative simplicity. They are traditionally defined via their marginal distributions, but a subset of DPPs called “L-ensembles” have tractable likelihoods and are thus particularly easy to work with. Indeed, in many applications, DPPs are more naturally defined based on the L-ensemble formulation rather than through the marginal kernel. The fact that not all DPPs are L-ensembles is unfortunate, but there is a unifying description. We introduce here extended L-ensembles, and show that all DPPs are extended L-ensembles (and vice versa). Extended L-ensembles have very simple likelihood functions, contain L-ensembles and projection DPPs as special cases. From a theoretical standpoint, they fix some pathologies in the usual formalism of DPPs, for instance, the fact that projection DPPs are not L-ensembles. From a practical standpoint, they extend the set of kernel functions that may be used to define DPPs: we show that conditional positive definite kernels are good candidates for defining DPPs, including DPPs that need no spatial scale parameter. Finally, extended L-ensembles are based on so-called “saddle-point matrices”, and we prove an extension of the Cauchy–Binet theorem for such matrices that may be of independent interest.

The elliptic Ginibre ensemble: A unifying approach to local and global statistics for higher dimensions

The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows us to interpolate between the rotationally invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a two-dimensional one-component Coulomb gas in a quadrupolar field at inverse temperature β = 2. Furthermore, it represents a determinantal point process in the complex plane with the corresponding kernel of planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. We provide a unifying approach to rigorously derive several known and new results of local and global spectral statistics, including in higher dimensions. First, we prove the global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici [Int. J. Mod. Phys. A 11, 941 (1996)]. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. In the Hermitian limit, there is a known correspondence between non-interacting fermions in a trap in d real dimensions Rd and the d-dimensional harmonic oscillator. We present a rigorous proof for the local d-dimensional bulk (sine) and edge (Airy) kernel first defined by Dean et al. [Europhys. Lett. 112, 60001 (2015)], complementing the recent results by Deleporte and Lambert [arXiv:2109.02121 (2021)]. Using the same relation to the d-dimensional harmonic oscillator in d complex dimensions Cd, we provide new local bulk and edge statistics at weak and strong non-Hermiticity, where the former interpolates between correlations in d real and d complex dimensions. For Cd with d = 1, this corresponds to non-interacting fermions in a rotating trap.

Fermion sampling made more efficient

Fermion sampling is to generate probability distribution of a many-body Slater-determinant wavefunction, which is termed "determinantal point process" in statistical analysis. For its inherently-embedded Pauli exclusion principle, its application reaches beyond simulating fermionic quantum many-body physics to constructing machine learning models for diversified datasets. Here we propose a fermion sampling algorithm, which has a polynomial time-complexity -- quadratic in the fermion number and linear in the system size. This algorithm is about 100% more efficient in computation time than the best known algorithms. In sampling the corresponding marginal distribution, our algorithm has a more drastic improvement, achieving a scaling advantage. We demonstrate its power on several test applications, including sampling fermions in a many-body system and a machine learning task of text summarization, and confirm its improved computation efficiency over other methods by counting floating-point operations.

A Data-Driven Optimization Method Considering Data Correlations for Optimal Power Flow Under Uncertainty

This paper introduces a data-driven optimization (DDO) method based on novel strategic sampling (SS) considering data correlations for multiperiod optimal power flow (OPF) considering energy storage devices under uncertainty (OPF-ESDUU) of uncertain renewable energy and power loads (UREPL). This DDO method depends only on the uncertainty samples to yield an optimal solution that satisfies a specific confidence level, which is effective because of two resounding learning algorithms: Bayesian hierarchical modeling (BHM) and determinantal point process (DPP). Considering both the local bus information and spatial correlations over all buses, BHM learns the convex approximation of AC power flow (CAACPF) more accurately than the existing learning methods, converting the originally non-convex OPF-ESDUU to a convex optimization problem. DPP considers the correlations between samples to find a small set of significant samples by measuring the relative weight of each sample using the random matrix theory, significantly decreasing the data samples required by the existing SS. The experimental analysis in IEEE test cases shows that after considering data correlations, 1) BHM learns CAACPF better with 13–90% accuracy improvement on average, compared with the existing learning methods, and 2) the proposed DDO performs more efficiently than the existing DDO as DPP-based SS boosts the sampling efficiency by 50% at least.

Deep Time Series Forecasting With Shape and Temporal Criteria.

This paper addresses the problem of multi-step time series forecasting for non-stationary signals that can present sudden changes. Current state-of-the-art deep learning forecasting methods, often trained with variants of the MSE, lack the ability to provide sharp predictions in deterministic and probabilistic contexts. To handle these challenges, we propose to incorporate shape and temporal criteria in the training objective of deep models. We define shape and temporal similarities and dissimilarities, based on a smooth relaxation of Dynamic Time Warping (DTW) and Temporal Distortion Index (TDI), that enable to build differentiable loss functions and positive semi-definite (PSD) kernels. With these tools, we introduce DILATE (DIstortion Loss including shApe and TimE), a new objective for deterministic forecasting, that explicitly incorporates two terms supporting precise shape and temporal change detection. For probabilistic forecasting, we introduce STRIPE++ (Shape and Time diverRsIty in Probabilistic for Ecasting), a framework for providing a set of sharp and diverse forecasts, where the structured shape and time diversity is enforced with a determinantal point process (DPP) diversity loss. Extensive experiments and ablations studies on synthetic and real-world datasets confirm the benefits of leveraging shape and time features in time series forecasting.

Sub-Poissonian Estimates for Exponential Moments of Additive Functionals over Pairs of Particles with Respect to Determinantal and Symplectic Pfaffian Point Processes Governed by Entire Functions

Sub-Poissonian Estimates for Exponential Moments of Additive Functionals over Pairs of Particles with Respect to Determinantal and Symplectic Pfaffian Point Processes Governed by Entire Functions

On the mean projection theorem for determinantal point processes

On the mean projection theorem for determinantal point processes

Wronskian structures of planar symplectic ensembles

We consider the eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble, which are known to form a Pfaffian point process in the plane. It was recently discovered that the limiting correlation kernel of the symplectic Ginibre ensemble in the vicinity of the real line can be expressed in a unified form of a Wronskian. We derive scaling limits for variations of the symplectic Ginibre ensemble and obtain such Wronskian structures for the associated universality classes. These include almost-Hermitian bulk/edge scaling limits of the elliptic symplectic Ginibre ensemble and edge scaling limits of the symplectic Ginibre ensemble with boundary confinement. Our proofs follow from the generalised Christoffel–Darboux formula for the former and from the Laplace method for the latter. Based on such a unified integrable structure of Wronskian form, we also provide an intimate relation between the function in the argument of the Wronskian in the symplectic symmetry class and the kernel in the complex symmetry class which form determinantal point processes in the plane.

Time–frequency recurrent transformer with diversity constraint for dense video captioning

Describing a long video using multiple sentences, i.e., dense video captioning, is a very challenging task. Existing methods neglect the important fact that the actions of several tempos (a.k.a., frequencies) evolve with the time in video, and do not well handle the phrase repetition issue. Therefore, we propose a Time-Frequency recurrent Transformer with Diversity constraint (TFTD) for dense video captioning. Its basic idea is to develop a time–frequency memory module, which not only stores the history of the past sentences and corresponding video segments to consider the temporal relations, but also models the motion dependency of action patterns with different frequencies. This contributes to producing more coherent sentences that well describe the video content. Moreover, we adopt the Determinantal Point Processes (DDP) for designing the diversity loss imposed to the objective function as constraint, such that the generated sentences are diverse with less redundancy. Extensive experiments on two benchmark datasets have verified the superior performance of our approach, e.g., it achieves 11.36, 16.56, 26.16, and 3.77, in terms of BLEU@4, METEOR, CIDEr-D, and R@4, respectively, on ActivityNet Captions (20,000 videos). Besides, TFTD outperforms the most competitive alternative by 8.0% on ActivityNet Captions and 9.8% on YouCookII (2,000 videos) in terms of coherence for human evaluation.

Truncations of Random Unitary Matrices Drawn from Hua-Pickrell Distribution

Let U be a random unitary matrix drawn from the Hua-Pickrell distribution \(\mu _{\textrm{U}(n+m)}^{(\delta )}\) on the unitary group \(\textrm{U}(n+m)\). We show that the eigenvalues of the truncated unitary matrix \([U_{i,j}]_{1\le i,j\le n}\) form a determinantal point process \(\mathscr {X}_n^{(m,\delta )}\) on the unit disc \(\mathbb {D}\) for any \(\delta \in \mathbb {C}\) satisfying \(\textrm{Re}\,\delta >-1/2\). We also prove that the limiting point process taken by \(n\rightarrow \infty \) of the determinantal point process \(\mathscr {X}_n^{(m,\delta )}\) is always \(\mathscr {X}^{[m]}\), independent of \(\delta \). Here \(\mathscr {X}^{[m]}\) is the determinantal point process on \(\mathbb {D}\) with weighted Bergman kernel $$\begin{aligned} \begin{aligned} K^{[m]}(z,w)=\frac{1}{(1-z{\overline{w}})^{m+1}} \end{aligned} \end{aligned}$$with respect to the reference measure \(d\mu ^{[m]}(z)=\frac{m}{\pi }(1-|z|)^{m-1}d\sigma (z)\), where \(d\sigma (z)\) is the Lebesgue measure on \(\mathbb {D}\).

Determinantal shot noise Cox processes

We present a new class of cluster point process models, which we call determinantal shot noise Cox processes (DSNCP), with repulsion between cluster centres. They are the special case of generalized shot noise Cox processes where the cluster centres are determinantal point processes. We establish various moment results and describe how these can be used to easily estimate unknown parameters in two particularly tractable cases, namely, when the offspring density is isotropic Gaussian and the kernel of the determinantal point process of cluster centres is Gaussian or like in a scaled Ginibre point process. Through a simulation study and the analysis of a real point pattern data set, we see that when modelling clustered point patterns, a much lower intensity of cluster centres may be needed in DSNCP models as compared to shot noise Cox processes.