Determinantal Point Processes Research Articles

Approximate Bayesian Computation and Model Assessment for Repulsive Spatial Point Processes

ABSTRACTIn many applications involving spatial point patterns, we find evidence of inhibition or repulsion. The most commonly used class of models for such settings are the Gibbs point processes. A recent alternative, at least to the statistical community, is the determinantal point process. Here, we examine model fitting and inference for both of these classes of processes in a Bayesian framework. While usual MCMC model fitting can be available, the algorithms are complex and are not always well behaved. We propose using approximate Bayesian computation (ABC) for such fitting. This approach becomes attractive because, though likelihoods are very challenging to work with for these processes, generation of realizations given parameter values is relatively straightforward. As a result, the ABC fitting approach is well-suited for these models. In addition, such simulation makes them well-suited for posterior predictive inference as well as for model assessment. We provide details for all of the above along with some simulation investigation and an illustrative analysis of a point pattern of tree data exhibiting repulsion. R code and datasets are included in the supplementary material.

Large Deviations for Gibbs Measures with Singular Hamiltonians and Emergence of K\xe4hler\u2013Einstein Metrics

In the present paper and the companion paper (Berman, Kähler–Einstein metrics, canonical random point processes and birational geometry. arXiv:1307.3634, 2015) a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on complex algebraic varieties X is introduced by sampling “temperature deformed” determinantal point processes. The main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper. As an application we show that the unique Kähler–Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point processes on X. In the companion paper (Berman in 2015) the extension to algebraic varieties X with positive Kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for Kähler–Einstein metrics with positive Ricci curvature is developed.

A palm hierarchy for determinantal point processes with the Bessel kernel

The main result of this note shows that Palm distributions of the determinantal point process governed by the Bessel kernel with parameter s are equivalent to the determinantal point process governed by the Bessel kernel with parameter s + 2. The Radon–Nikodym derivative is explicitly computed as a multiplicative functional on the space of configurations.

The ASEP and Determinantal Point Processes

We introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line, with correlation kernels defined via spectral projections for the associated Jacobi matrices. For classical weights, we show how such ensembles arise as limits of various hypergeometric orthogonal polynomials ensembles. We then prove that the q-Laplace transform of the height function of the ASEP with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete Laguerre ensemble --- a member of the new family. This allows us to obtain the large time asymptotics of the ASEP in three limit regimes: (a) for finitely many rightmost particles; (b) GUE Tracy-Widom asymptotics of the height function; (c) KPZ asymptotics of the height function for the ASEP with weak asymmetry. We also give similar results for two instances of the stochastic six vertex model in a quadrant. The proofs are based on limit transitions for the corresponding determinantal point processes.

Linear rigidity of stationary stochastic processes

We consider stationary stochastic processes $\{X_{n}:n\in \mathbb{Z}\}$ such that $X_{0}$ lies in the closed linear span of $\{X_{n}:n\neq 0\}$; following Ghosh and Peres, we call such processes linearly rigid. Using a criterion of Kolmogorov, we show that it suffices, for a stationary stochastic process to be linearly rigid, that the spectral density vanishes at zero and belongs to the Zygmund class $\unicode[STIX]{x1D6EC}_{\ast }(1)$. We next give a sufficient condition for stationary determinantal point processes on $\mathbb{Z}$ and on $\mathbb{R}$ to be linearly rigid. Finally, we show that the determinantal point process on $\mathbb{R}^{2}$ induced by a tensor square of Dyson sine kernels is not linearly rigid.

The Weyl–Heisenberg ensemble: hyperuniformity and higher Landau levels

Weyl–Heisenberg ensembles are a class of determinantal point processes associated with the Schrödinger representation of the Heisenberg group. Hyperuniformity characterizes a state of matter for which (scaled) density fluctuations diminish towards zero at the largest length scales. We will prove that Weyl–Heisenberg ensembles are hyperuniform. Weyl–Heisenberg ensembles include as a special case a multi-layer extension of the Ginibre ensemble modeling the distribution of electrons in higher Landau levels, which has recently been object of study in the realm of the Ginibre-type ensembles associated with polyanalytic functions. In addition, the family of Weyl–Heisenberg ensembles includes new structurally anisotropic processes, where point-statistics depend on the different spatial directions, and thus provide a first means to study directional hyperuniformity.

Conditional measures of generalized Ginibre point processes

The main result of this paper is that conditional measures of generalized Ginibre point processes, with respect to the configuration in the complement of a bounded open subset on C, are orthogonal polynomial ensembles with weights found explicitly. An especially simple formula for conditional measures is obtained in the particular case of radially-symmetric determinantal point processes, including the classical Ginibre point process.

Some Recent Developments in Statistics for Spatial Point Patterns

This article reviews developments in statistics for spatial point processes obtained within roughly the past decade. These developments include new classes of spatial point process models such as determinantal point processes, models incorporating both regularity and aggregation, and models where points are randomly distributed around latent geometric structures. Regarding parametric inference, the main focus is on various types of estimating functions derived from so-called innovation measures. Optimality of such estimating functions is discussed, as well as computational issues. Maximum likelihood inference for determinantal point processes and Bayesian inference are also briefly considered. Concerning nonparametric inference, we consider extensions of functional summary statistics to the case of inhomogeneous point processes as well as new approaches to simulation-based inference.

Rigid stationary determinantal processes in non-Archimedean fields

Let $F$ be a non-discrete non-Archimedean local field. For any subset $S\subset F$ with finite Haar measure, there is a stationary determinantal point process on $F$ with correlation kernel $\widehat{\mathbb{1}}_{S}(x-y)$, where $\widehat{\mathbb{1}}_{S}$ is the Fourier transform of the indicator function $\mathbb{1}_{S}$. In this note, we give a geometrical condition on the subset $S$, such that the associated determinantal point process is rigid in the sense of Ghosh and Peres. Our geometrical condition is very different from the Euclidean case.

Low-Rank Factorization of Determinantal Point Processes

Determinantal point processes (DPPs) have garnered attention as an elegant probabilistic model of set diversity. They are useful for a number of subset selection tasks, including product recommendation. DPPs are parametrized by a positive semi-definite kernel matrix. In this work we present a new method for learning the DPP kernel from observed data using a low-rank factorization of this kernel. We show that this low-rank factorization enables a learning algorithm that is nearly an order of magnitude faster than previous approaches, while also providing for a method for computing product recommendation predictions that is far faster (up to 20x faster or more for large item catalogs) than previous techniques that involve a full-rank DPP kernel. Furthermore, we show that our method provides equivalent or sometimes better test log-likelihood than prior full-rank DPP approaches.

Determinantal Point Processes Associated with Hilbert Spaces of Holomorphic Functions

We study determinantal point processes on $\mathbb{C}$ induced by the reproducing kernels of generalized Fock spaces as well as those on the unit disc $\mathbb{D}$ induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures $\textit{of the same order}$ are equivalent. The Radon-Nikodym derivatives are computed explicitly using regularized multiplicative functionals. We also show that these determinantal point processes are rigid in the sense of Ghosh and Peres, hence reduced Palm measures $\textit{of different orders}$ are singular. In the second case, we show that all reduced Palm measures, $\textit{of all orders}$, are equivalent. The Radon-Nikodym derivatives are computed using regularized multiplicative functionals associated with certain Blaschke products. The quasi-invariance of these determinantal point processes under the group of diffeomorphisms with compact supports follows as a corollary.

Infinite random matrices & ergodic decomposition of finite and infinite Hua–Pickrell measures

The ergodic decomposition of a family of Hua–Pickrell measures on the space of infinite Hermitian matrices is studied. By combining previous results of Borodin–Olshanski and our new results, we obtain the first complete description of the ergodic decomposition of Hua–Pickrell measures. First, we show that the ergodic components of any Hua–Pickrell probability measure have no Gaussian factors. Secondly, we show that the sequence of asymptotic eigenvalues of Hua–Pickrell random matrices is balanced in a certain sense and has a “principal value” which coincides with the parameter that reflects the presence of Dirac factor in an ergodic component. This allows us to identify the ergodic decomposition of any Hua–Pickrell probability with a certain determinantal point process with hypergeometric kernel as introduced by Borodin–Olshanski. Finally, we extend the aforesaid results to the case of infinite Hua–Pickrell measures. By using the theory of infinite determinantal measures recently introduced by A.I. Bufetov, we are able to identify the ergodic decomposition of Hua–Pickrell infinite measure with a certain infinite determinantal measure. This resolves a problem of Borodin and Olshanski.

Quasi-symmetries and rigidity for determinantal point processes associated with de Branges spaces

In this note, we show that determinantal point processes on the real line corresponding to de Branges spaces of entire functions are rigid in the sense of Ghosh-Peres and, under certain additional assumptions, quasi-invariant under the group of diffeomorphisms of the line with compact support.

On a few statistical applications of determinantal point processes

Determinantal point processes (DPPs) are a repulsive distribution over configurations of points. The 2016 conference Journees Modelisation Aleatoire et Statistique (MAS) of the French society for applied and industrial mathematics (SMAI) featured a session on statistical applications of DPPs. This paper gathers contributions by the speakers and the organizer of the session.

Discovering Diverse Subset for Unsupervised Hyperspectral Band Selection.

Band selection, as a special case of the feature selection problem, tries to remove redundant bands and select a few important bands to represent the whole image cube. This has attracted much attention, since the selected bands provide discriminative information for further applications and reduce the computational burden. Though hyperspectral band selection has gained rapid development in recent years, it is still a challenging task because of the following requirements: 1) an effective model can capture the underlying relations between different high-dimensional spectral bands; 2) a fast and robust measure function can adapt to general hyperspectral tasks; and 3) an efficient search strategy can find the desired selected bands in reasonable computational time. To satisfy these requirements, a multigraph determinantal point process (MDPP) model is proposed to capture the full structure between different bands and efficiently find the optimal band subset in extensive hyperspectral applications. There are three main contributions: 1) graphical model is naturally transferred to address band selection problem by the proposed MDPP; 2) multiple graphs are designed to capture the intrinsic relationships between hyperspectral bands; and 3) mixture DPP is proposed to model the multiple dependencies in the proposed multiple graphs, and offers an efficient search strategy to select the optimal bands. To verify the superiority of the proposed method, experiments have been conducted on three hyperspectral applications, such as hyperspectral classification, anomaly detection, and target detection. The reliability of the proposed method in generic hyperspectral tasks is experimentally proved on four real-world hyperspectral data sets.

Noninteracting fermions at finite temperature in ad-dimensional trap: Universal correlations

We study a system of $N$ non-interacting spin-less fermions trapped in a confining potential, in arbitrary dimensions $d$ and arbitrary temperature $T$. The presence of the trap introduces an edge where the average density of fermions vanishes. Far from the edge, near the center of the trap (the so called "bulk regime"), physical properties of the fermions have traditionally been understood using the Local Density Approximation. However, this approximation drastically fails near the edge where the density vanishes. In this paper we show that, even near the edge, novel universal properties emerge, independently of the details of the confining potential. We show that for large $N$, these fermions in a confining trap, in arbitrary dimensions and at finite temperature, form a determinantal point process. As a result, any $n$-point correlation function can be expressed as an $n \times n$ determinant whose entry is called the kernel. Near the edge, we derive the large $N$ scaling form of the kernels. In $d=1$ and $T=0$, this reduces to the so called Airy kernel, that appears in the Gaussian Unitary Ensemble (GUE) of random matrix theory. In $d=1$ and $T>0$ we show a remarkable connection between our kernel and the one appearing in the $1+1$-dimensional Kardar-Parisi-Zhang equation at finite time. Consequently our result provides a finite $T$ generalization of the Tracy-Widom distribution, that describes the fluctuations of the rightmost fermion at $T=0$. In $d>1$ and $T \geq 0$, while the connection to GUE no longer holds, the process is still determinantal whose analysis provides a new class of kernels, generalizing the $1d$ Airy kernel at $T=0$ obtained in random matrix theory. Some of our finite temperature results should be testable in present-day cold atom experiments, most notably our detailed predictions for the temperature dependence of the fluctuations near the edge.

Contrast Estimation for Parametric Stationary Determinantal Point Processes

AbstractWe study minimum contrast estimation for parametric stationary determinantal point processes. These processes form a useful class of models for repulsive (or regular, or inhibitive) point patterns and are already applied in numerous statistical applications. Our main focus is on minimum contrast methods based on the Ripley's K‐function or on the pair correlation function. Strong consistency and asymptotic normality of theses procedures are proved under general conditions that only concern the existence of the process and its regularity with respect to the parameters. A key ingredient of the proofs is the recently established Brillinger mixing property of stationary determinantal point processes. This work may be viewed as a complement to the study of Y. Guan and M. Sherman who establish the same kind of asymptotic properties for a large class of Cox processes, which in turn are models for clustering (or aggregation).

Bulk and soft-edge universality for singular values of products of Ginibre random matrices

Il a été démontré par Akemann, Ipsen et Kieburg que les valeurs singulières carrées de produits de $M$ matrices aléatoires rectangulaires composées de variables aléatoires complexes gaussiennes, suivent un processus ponctuel déterminantal, avec un noyau de corrélation qui admet une représentation en fonction des fonctions G de Meijer. Nous démontrons l’universalité des statistiques locales des valeurs singulières carrées, c’est-à-dire, l’universalité dans le bulk donné par le noyau sinus et l’universalité au bord donné par le noyau Airy. La preuve est basée sur l’analyse asymptotique de la représentation du noyau de corrélation comme une double intégrale de contour. Notre stratégie peut être généralisée à d’autres modèles de produits de matrices aléatoires introduits récemment, afin d’obtenir des résultats d’universalité similaires. Deux autres exemples sont considérés, le premier étant le produit de $M$ matrices de Ginibre et l’inverse de $K$ matrices de Ginibre, étudié par Forrester, le deuxième étant le produit de $M-1$ matrices de Ginibre avec une troncation d’une matrice unitaire, considéré par Kuijlaars et Stivigny.

Moments Match between the KPZ Equation and the Airy Point Process

The results of Amir-Corwin-Quastel, Calabrese-Le Doussal-Rosso, Dotsenko, and Sasamoto-Spohn imply that the one-point distribution of the solution of the KPZ equation with the narrow wedge initial condition coincides with that for a multiplicative statistics of the Airy determinantal random point process. Taking Taylor coefficients of the two sides yields moment identities. We provide a simple direct proof of those via a combinatorial match of their multivariate integral representations.

Hole Probabilities for Finite and Infinite Ginibre Ensembles:

We study the hole probabilities of the infinite Ginibre ensemble ${\mathcal X}_{\infty}$, a determinantal point process on the complex plane with the kernel $\mathbb K(z,w)= \frac{1}{\pi}e^{z\bar w-\frac{1}{2}|z|^2-\frac{1}{2}|w|^2}$ with respect to the Lebesgue measure on the complex plane. Let $U$ be an open subset of open unit disk $\mathbb D$ and ${\mathcal X}_{\infty}(rU)$ denote the number of points of ${\mathcal X}_{\infty}$ that fall in $rU$. Then, under some conditions on $U$, we show that $$ \lim_{r\to \infty}\frac{1}{r^4}\log\mathbb P[\mathcal X_{\infty}(rU)=0]=R_{\emptyset}-R_{U}, $$ where $\emptyset$ is the empty set and $$ R_U:=\inf_{\mu\in \mathcal P(U^c)}\left\{\iint \log{\frac{1}{|z-w|}}d\mu(z)d\mu(w)+\int |z|^2d\mu(z) \right\}, $$ $\mathcal P(U^c)$ is the space of all compactly supported probability measures with support in $U^c$. Using potential theory, we give an explicit formula for $R_U$, the minimum possible energy of a probability measure compactly supported on $U^c$ under logarithmic potential with a quadratic external field. Moreover, we calculate $R_U$ explicitly for some special sets like annulus, cardioid, ellipse, equilateral triangle and half disk.