Determinantal Point Process Models Research Articles

A Learning Approach for Efficient Multicast Beamforming Based on Determinantal Point Process

The problem of single-group multicast beamforming (SMBF) is well-known NP-hard. It motivates the pursuit of computationally efficient near-optimal solutions. Due to multicasting, the multicast group is bottlenecked by the user(s) with the minimum received signal-to-noise ratio (SNR). This paper provides an in-depth interpretation of the SMBF problem from the multicasting point of view and proposes to solve it in two steps: i) select the bottlenecking users by a machine learning approach based on determinantal point process (DPP), and ii) design the beamformer for the selected users. The DPP model jointly considers the magnitudes and directions of users’ channel vectors, and thus enables an efficient selection of the bottlenecking users. Moreover, for a specific channel model, the DPP model is only associated with network size and each takes a one-off training cost, thus can be used as a codebook. The proposed DPP-based subset selection is incorporated adaptively into two fast beamforming algorithms, i.e., the QR decomposition algorithm and the successive beamforming (SB) algorithm. They specifically design the beamformers for the selected users by leveraging channel orthogonalization therein. Numerical results demonstrate the superiority of the proposed QR-DPP and SB-DPP algorithms in terms of the performance-complexity compromise and their robustness to different scenarios.

Diversity-augmented intrinsic motivation for deep reinforcement learning

In many real-world problems, reward signals received by agents are delayed or sparse, which makes it challenging to train a reinforcement learning (RL) agent. An intrinsic reward signal can help an agent to explore such environments in the quest for novel states. In this work, we propose a general end-to-end diversity-augmented intrinsic motivation for deep reinforcement learning which encourages the agent to explore new states and automatically provides denser rewards. Specifically, we measure the diversity of adjacent states under a model of state sequences based on determinantal point process (DPP); this is coupled with a straight-through gradient estimator to enable end-to-end differentiability. The proposed approach is comprehensively evaluated on the MuJoCo and the Arcade Learning Environments (Atari and SuperMarioBros). The experiments show that an intrinsic reward based on the diversity measure derived from the DPP model accelerates the early stages of training in Atari games and SuperMarioBros. In MuJoCo, the approach improves on prior techniques for tasks using the standard reward setting, and achieves the state-of-the-art performance on 12 out of 15 tasks containing delayed rewards.

A retrieval-based approach for diverse and image-specific adversary selection

While deep neural network-based models have demonstrated compelling performance on various tasks in computer vision and other fields, they have been found to be vulnerable to adversarial attacks. Particularly, deep convolutional neural network (CNN)-based models can be easily fooled by adding a small quasi-imperceptible perturbation to the input, thus resulting in significant drop in prediction accuracies. While most of the previous works have focused on generating one adversary/perturbation per model, it was recently shown that it is possible to learn a continuous distribution over adversarial perturbations for a model. Building upon this work, in this paper, we propose a new technique for image-specific adversary selection and treat it as a retrieval task. The proposed technique utilizes a learned model that ranks the perturbations in a given set of perturbations based on their ability to fool with respect to a given sample. This model is a conditional determinantal point process model that also explicitly induces diversity among the retrieved perturbations. We conduct experiments on the ImageNet dataset using four popular deep CNN image classification models, and demonstrate that the proposed method consistently achieves state-of-the-art fooling rates.

Reach of repulsion for determinantal point processes in high dimensions

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.

Determinantal point process models on the sphere

We consider determinantal point processes on the $d$-dimensional unit sphere $\mathbb{S}^{d}$. These are finite point processes exhibiting repulsiveness and with moment properties determined by a certain determinant whose entries are specified by a so-called kernel which we assume is a complex covariance function defined on $\mathbb{S}^{d}\times\mathbb{S}^{d}$. We review the appealing properties of such processes, including their specific moment properties, density expressions and simulation procedures. Particularly, we characterize and construct isotropic DPPs models on $\mathbb{S}^{d}$, where it becomes essential to specify the eigenvalues and eigenfunctions in a spectral representation for the kernel, and we figure out how repulsive isotropic DPPs can be. Moreover, we discuss the shortcomings of adapting existing models for isotropic covariance functions and consider strategies for developing new models, including a useful spectral approach.

Statistical Modeling and Probabilistic Analysis of Cellular Networks With Determinantal Point Processes

Although the Poisson point process (PPP) has been widely used to model base station (BS) locations in cellular networks, it is an idealized model that neglects the spatial correlation among BSs. This paper proposes the use of the determinantal point process (DPP) to take into account these correlations, in particular the repulsiveness among macro BS locations. DPPs are demonstrated to be analytically tractable by leveraging several unique computational properties. Specifically, we show that the empty space function, the nearest neighbor function, the mean interference, and the signal-to-interference ratio (SIR) distribution have explicit analytical representations and can be numerically evaluated for cellular networks with DPP-configured BSs. In addition, the modeling accuracy of DPPs is investigated by fitting three DPP models to real BS location data sets from two major U.S. cities. Using hypothesis testing for various performance metrics of interest, we show that these fitted DPPs are significantly more accurate than popular choices such as the PPP and the perturbed hexagonal grid model.

Determinantal Point Process Models and Statistical Inference

Summary Statistical models and methods for determinantal point processes (DPPs) seem largely unexplored. We demonstrate that DPPs provide useful models for the description of spatial point pattern data sets where nearby points repel each other. Such data are usually modelled by Gibbs point processes, where the likelihood and moment expressions are intractable and simulations are time consuming. We exploit the appealing probabilistic properties of DPPs to develop parametric models, where the likelihood and moment expressions can be easily evaluated and realizations can be quickly simulated. We discuss how statistical inference is conducted by using the likelihood or moment properties of DPP models, and we provide freely available software for simulation and statistical inference.