Generalized Determinantal Point Processes: The Linear Case

A determinantal point process (DPP) over a universe $\{1,\ldots,m\}$ with respect to an $m \times m$ positive semidefinite matrix $L$ is a probability distribution where the probability of a subset $S \subseteq \{1,\ldots,m\}$ is proportional to the determinant of the principal minor of $L$ corresponding to $S.$ DPPs encapsulate a wide variety of known distributions and appear naturally (and surprisingly) in a wide variety of areas such as physics, mathematics and computer science. Several applications that use DPPs rely on the fact that they are computationally tractable -- i.e., there are algorithms for sampling from DPPs efficiently. Recently, there is growing interest in studying a generalization of DPPs in which the support of the distribution is a restricted family B of subsets of $\{1,2,\ldots, m\}$. Mathematically, these distributions, which we call generalized DPPs, include the well-studied hardcore distributions as special cases (when $L$ is diagonal). In applications, they can be used to refine models based on DPPs by imposing combinatorial constraints on the support of the distribution. In this paper we take first steps in a systematic study of computational questions concerning generalized DPPs. We introduce a natural class of linear families: roughly, a family B is said to be linear if there is a collection of $p$ linear forms that all elements of B satisfy. Important special cases of linear families are all sets of cardinality $k$ -- giving rise to $k$-DPPs -- and, more generally, partition matroids. On the positive side, we prove that, when $p$ is a constant, there is an efficient, exact sampling algorithm for linear DPPs. We complement these results by proving that, when $p$ is large, the computational problem related to such DPPs becomes $\#$P-hard. Our proof techniques rely and build on the interplay between polynomials and probability distributions.