# Computational Complexity of Normalizing Constants for the Product of Determinantal Point Processes

We consider the product of determinantal point processes (DPPs), a point process whose probability mass is proportional to the product of principal minors of multiple matrices, as a natural, promising generalization of DPPs. We study the computational complexity of computing its normalizing constant, which is among the most essential probabilistic inference tasks. Our complexity-theoretic results (almost) rule out the existence of efficient algorithms for this task unless the input matrices are forced to have favorable structures. In particular, we prove the following: (1) Computing $\sum_S\det({\bf A}_{S,S})^p$ exactly for every (fixed) positive even integer $p$ is UP-hard and Mod$_3$P-hard, which gives a negative answer to an open question posed by Kulesza and Taskar. (2) $\sum_S\det({\bf A}_{S,S})\det({\bf B}_{S,S})\det({\bf C}_{S,S})$ is NP-hard to approximate within a factor of $2^{O(|I|^{1-\epsilon})}$ or $2^{O(n^{1/\epsilon})}$ for any $\epsilon>0$, where $|I|$ is the input size and $n$ is the order of the input matrix. This result is stronger than the #P-hardness for the case of two matrices derived by Gillenwater. (3) There exists a $k^{O(k)}n^{O(1)}$-time algorithm for computing $\sum_S\det({\bf A}_{S,S})\det({\bf B}_{S,S})$, where $k$ is the maximum rank of $\bf A$ and $\bf B$ or the treewidth of the graph formed by nonzero entries of $\bf A$ and $\bf B$. Such parameterized algorithms are said to be fixed-parameter tractable. These results can be extended to the fixed-size case. Further, we present two applications of fixed-parameter tractable algorithms given a matrix $\bf A$ of treewidth $w$: (4) We can compute a $2^{\frac{n}{2p-1}}$-approximation to $\sum_S\det({\bf A}_{S,S})^p$ for any fractional number $p>1$ in $w^{O(wp)}n^{O(1)}$ time. (5) We can find a $2^{\sqrt n}$-approximation to unconstrained MAP inference in $w^{O(w\sqrt n)}n^{O(1)}$ time.

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