A METRIC APPROACH TOWARD POINT PROCESS DIVERGENCE Sohan Seth, Austin J. Brockmeier, Jos´ e C. Pr´ ıncipe Electrical and Computer Engineering University of Florida, Gainesville, Florida {sohan,ajbrockmeier,principe}@cnel.ufl.edu ABSTRACT Estimating divergence between two point processes, i.e. probability laws on the space of spike trains, is an essential tool in many computational neuroscience applications, such as change detection and neural coding. However, the prob- lem of estimating divergence, although well studied in the Euclidean space, has seldom been addressed in a more gen- eral setting. Since the space of spike trains can be viewed as a metric space, we address the problem of estimating Jensen- Shannon divergence in a metric space using a nearest neigh- bor based approach. We empirically demonstrate the validity of the proposed estimator, and compare it against other avail- able methods in the context of two-sample problem. Index Terms— Divergence, metric space, point process, nearest neighbor, hypothesis testing 1. INTRODUCTION In the neuroscience literature, it is a well accepted fact that neurons communicate via sequence of action potentials or spikes. This sequence is formally known as a neural spike train, and the probability law over the set of spike trains is for- mally known as a point process. Many computational neuro- science applications such as change detection, can be framed as a two-sample problem, that requires evaluating a suitable divergence measure between two point processes; where a di- vergence measure is defined as a non-negative statistic of two probability laws that attains zero value if and only if the prob- ability laws are the same. However, estimating divergence between two point processes is not trivial since the space of spike trains lacks basic algebraic and topological structures. Recently, [5] and [7] have proposed two conceptually different approaches for assessing divergence between two point processes: [5] has addressed the problem of estimating Hellinger distance between two point processes, whereas [7] has extended the Kolmogorov-Smirnov and the Cram´ er-von- Mises tests to the space of spike trains. Both these approaches rely on representing the space of spike trains as a direct sum The work has been partially supported by NSF grant ECCS-0856441 and DARPA grant N66001-10-C-2008. The authors thank Dr. Il Park for valuable discussions at an early stage of this project. of Euclidean spaces, called strata, and evaluating the di- vergence separately in each stratum. Although theoretically sound and computationally efficient, these approaches have a serious drawback; they do not allow any interaction between two spike trains from two different strata. Therefore, these approaches suffer in estimation since the strata are created based on the spike counts, and given a finite number of spike trains, it becomes hard to populate each stratum sufficiently, especially when the underlying point process has a relatively flat count distribution. This issue restricts the applicability of these approaches. In this paper, we explore an alternate method of estimating divergence that do not require stratifi- cation. In an Euclidean space, the problem of estimating diver- gence can be addressed in several ways, e.g. using the empiri- cal cumulative distribution function or the empirical Randon- Nikodym derivative [2, 8, 10]. These approaches, however, cannot be readily applied to a more general space. The former approach requires the underlying space to have an order. On the other hand, the latter approach can be materialized in sev- eral ways including using a nearest neighbor based approach [10]. We aim at extending this approach since, conceptually, it only requires the underlying space to have a metric. Since, recent research in the field of neuroscience have shown that the space of spike trains can be treated as a metric space [9], it allows the proposed estimator to be used in the context of estimating divergence between two point processes. The rest of the paper is organized as follows. In section 2 we describe the difficulties of extending the traditional meth- ods of estimating divergence to a more general space, and then propose a method of estimating Jensen-Shannon diver- gence in a metric space. We choose this divergence since it is widely used as a metric in the space of probability mea- sures [4]. In section 3 we discuss the effect of the choice of metric on the convergence of the estimated divergence value, and present three simulated hypothesis testing experiments to demonstrate the validity and the drawbacks of the current view of estimating divergence. In section 4 we conclude the paper with some formal discussion on the pros and cons of the proposed method, and some guidelines of future work.