Graphical structure of conditional independencies in determinantal point processes

Determinantal point process have recently been used as models in machine learning and this has raised questions regarding the characterizations of conditional independence. In this paper we investigate characterizations of conditional independence. We describe some conditional independencies through the conditions on the kernel of a determinantal point process, and show many can be obtained using the graph induced by a kernel of the L-ensemble. In recent years there have been several machine learning papers about the applications of determinantal point processes (DPP's) [4], [7], [8], [9]... An overview of theory, recent applications and problems in learning DPP's is given in a recent extensive survey [6] by Kulesza and Taskar. In a private communication with Ben Taskar, one of the questions from survey [6] (see §7.3), that remains for future research, was brought to author's attention: - Is there a simple characterization of the conditional independence relations encoded by a DPP? This question arises naturally having in mind conditional independence structure models (see [12]), such as graphical models (see [11]) that are often used. It turns out that, from the mathematical view point, elegant characterizations, similar to those in graphical models, exist. This paper provides two (main) characterizations: - the block in a Schur complement of the kernel has to be a 0-block (Theorem 16, Proposition 17); - we can use the structure of the graph induced by the kernel of the L-ensemble to read many conditional independencies in the process (Theorem 28, Proposition 30).