EQUIVARIANT NORMALIZING FLOWS FOR POINT PROCESSES AND SETS Anonymous authors Paper under double-blind review

Abstract

A point process describes how random sets of exchangeable points are generated. The points usually influence the positions of each other via attractive and repulsive forces. To model this behavior, it is enough to transform the samples from the uniform process with a sufficiently complex equivariant function. However, learning the parameters of the resulting process is challenging since the likelihood is hard to estimate and often intractable. This leads us to our proposed model -CONFET. Based on continuous normalizing flows, it allows arbitrary interactions between points while having tractable likelihood. Experiments on various real and synthetic datasets show the improved performance of our new scalable approach.

1. INTRODUCTION

Many domains contain unordered data with a variable number of elements. The lack of ordering, also known as exchangeability, can be found in locations of cellular stations, locations of trees in a forest, point clouds, items in a shopping cart etc. This kind of data is represented with sets that are randomly generated from some underlying process that we wish to uncover. We choose to model this with spatial point processes, generative models whose realizations are sets of points. Perhaps the simplest non-trivial model is an inhomogeneous Poisson process. The locations of the points are assumed to be generated i.i.d. from some density (Chiu et al., 2013) . By simply modeling this density we can evaluate the likelihood and draw samples. We can do this easily with normalizing flows (Germain et al., 2015) . The process is then defined with a transformation of samples from a simple distribution to samples in the target distribution. However, the i.i.d. property is often wrong because the presence of one object will influence the distribution of the others. For example, short trees grow near each other, but are inhibited by taller trees (Ogata & Tanemura, 1985) . As an example we can generate points on the interval (0, 1) in the following way: we first flip a coin to decide whether to sample inside or outside of the interval (1/4, 3/4); then sample two points x 1 , x 2 uniformly on the chosen subset. Although the marginals p(x 1 ) and p(x 2 ) are uniformly distributed, knowing the position of one point gives us information about the other. Therefore, we should not model this process as if the points are independent, but model the joint distribution p(x 1 , x 2 ), with a constraint that p is symmetric to permutations (Figure 1 ). Unfortunately, when we include interactions between the points, the problem becomes significantly harder because the likelihood is intractable (Besag, 1975) . 



Figure 1: (Left) A symmetric density and its samples, see details in text. (Right) Illustration of our approach, going from the uniform to the target process with a continuous normalizing flow.

