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Computational complexity of normalizing constants for the product of determinantal point processes.
- Source :
-
Theoretical Computer Science . May2024, Vol. 997, pN.PAG-N.PAG. 1p. - Publication Year :
- 2024
-
Abstract
- 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: • Computing ∑ S det (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 [51]. • ∑ S det (A S , S) det (B S , S) det (C S , S) is NP -hard to approximate within a factor of 2 O (| I | 1 − ε) or 2 O (n 1 / ε) for any ε > 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 [36]. • There exists a k O (k) n O (1) -time algorithm for computing ∑ S det (A S , S) det (B S , S) , where k is the maximum rank of A and B or the treewidth of the graph formed by nonzero entries of A and 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 A of treewidth w : • We can compute a 2 n 2 p − 1 -approximation to ∑ S det (A S , S) p for any fractional number p > 1 in w O (w p) n O (1) time. • We can find a 2 n -approximation to unconstrained maximum a posteriori inference in w O (w n) n O (1) time. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 03043975
- Volume :
- 997
- Database :
- Academic Search Index
- Journal :
- Theoretical Computer Science
- Publication Type :
- Academic Journal
- Accession number :
- 176537749
- Full Text :
- https://doi.org/10.1016/j.tcs.2024.114513