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

Jul 12, 2020

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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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About ICML 2020

The International Conference on Machine Learning (ICML) is the premier gathering of professionals dedicated to the advancement of the branch of artificial intelligence known as machine learning. ICML is globally renowned for presenting and publishing cutting-edge research on all aspects of machine learning used in closely related areas like artificial intelligence, statistics and data science, as well as important application areas such as machine vision, computational biology, speech recognition, and robotics. ICML is one of the fastest growing artificial intelligence conferences in the world. Participants at ICML span a wide range of backgrounds, from academic and industrial researchers, to entrepreneurs and engineers, to graduate students and postdocs.

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