On the (In)tractability of Computing Normalizing Constants for the Product of Determinantal Point Processes

Jul 12, 2020



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 input matrices are forced to have favorable structures. In particular, we prove the following: (1) Computing ∑_S(A_S,S)^p exactly for every (fixed) positive even integer p is UP-hard and Mod3P-hard, which gives a negative answer to an open question posed by Kulesza and Taskar (2012). (2) ∑_S(A_S,S) (B_S,S) (C_S,S) is NP-hard to approximate within a factor of 2^O(|I|^1-ϵ) for any ϵ > 0, where |I| is the input size. This result is stronger than #P-hardness for the case of two matrices by Gillenwater (2014). (3) There exists a k^O(k) |I|^O(1)-time algorithm for computing ∑_S(A_S,S) (B_S,S), where k is “the maximum rank of A and B” or “the treewidth of the graph formed by non-zero entries of A and B.” Such parameterized algorithms are said to be fixed-parameter tractable.


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