Combinatorial Determinantal Point Processes

Determinantal Point Processes (DPPs) are probabilistic models of repulsion that originate in quantum physics and random matrix theory and have been of recent interest in computer science. DPPs define distributions over subsets of a given ground set, and exhibit interesting properties such as negative correlation. When applied to kernel methods in machine learning, DPPs give rise to an efficient algorithm to select a small, diverse sample of the given data. Kulesza and Taskar [KT12] posed a natural open question: Can we sample from DPPs when there are additional constraints on the allowable subsets? In this paper, we study the complexity of sampling from DPPs over combinatorially-constrained families of subsets and present several connections and applications. We start by showing that it is at least as hard to sample from combinatorial DPPs as it is to compute the mixed discriminant of a tuple of positive semidefinite matrices. Subsequently, we give a polynomial time algorithm for sampling from combinatorial DPPs with a constant number of linear constraints; thus, we make significant progress towards answering the question of [KT12]. These results can lead to several non-trivial applications; e.g., we show how to derandomize a result of Nikolov and Singh [NS16] for maximizing subdeterminants under (a constant number of) partition constraints and -- motivated by making the [MSS13] proof for the Kadison-Singer problem algorithmic -- give an algorithm to compute the higher-order coefficients of mixed characteristic polynomials.