Computational complexity of normalizing constants for the product of determinantal point processes

Tatsuya Matsuoka, Naoto Ohsaka

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 ∑Sdet⁡(AS,S)p<math><msub is="true"><mrow is="true"><mo is="true">∑</mo></mrow><mrow is="true"><mi is="true">S</mi></mrow></msub><mi mathvariant="normal" is="true">det</mi><mo is="true">⁡</mo><msup is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">A</mi></mrow><mrow is="true"><mi is="true">S</mi><mo is="true">,</mo><mi is="true">S</mi></mrow></msub><mo stretchy="false" is="true">)</mo></mrow><mrow is="true"><mi is="true">p</mi></mrow></msup></math> 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 [51].•∑Sdet⁡(AS,S)det⁡(BS,S)det⁡(CS,S)<math><msub is="true"><mrow is="true"><mo is="true">∑</mo></mrow><mrow is="true"><mi is="true">S</mi></mrow></msub><mi mathvariant="normal" is="true">det</mi><mo is="true">⁡</mo><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">A</mi></mrow><mrow is="true"><mi is="true">S</mi><mo is="true">,</mo><mi is="true">S</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mi mathvariant="normal" is="true">det</mi><mo is="true">⁡</mo><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">B</mi></mrow><mrow is="true"><mi is="true">S</mi><mo is="true">,</mo><mi is="true">S</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mi mathvariant="normal" is="true">det</mi><mo is="true">⁡</mo><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">C</mi></mrow><mrow is="true"><mi is="true">S</mi><mo is="true">,</mo><mi is="true">S</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math> is NP-hard to approximate within a factor of 2O(|I|1−ε)<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mo stretchy="false" is="true">|</mo><mi mathvariant="script" is="true">I</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mn is="true">1</mn><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ε</mi></mrow></msup><mo stretchy="false" is="true">)</mo></mrow></msup></math> or 2O(n1/ε)<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">1</mn><mo stretchy="false" is="true">/</mo><mi is="true">ε</mi></mrow></msup><mo stretchy="false" is="true">)</mo></mrow></msup></math> for any ε>0<math><mi is="true">ε</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></math>, where |I|<math><mo stretchy="false" is="true">|</mo><mi mathvariant="script" is="true">I</mi><mo stretchy="false" is="true">|</mo></math> 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 kO(k)nO(1)<math><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math>-time algorithm for computing ∑Sdet⁡(AS,S)det⁡(BS,S)<math><msub is="true"><mrow is="true"><mo is="true">∑</mo></mrow><mrow is="true"><mi is="true">S</mi></mrow></msub><mi mathvariant="normal" is="true">det</mi><mo is="true">⁡</mo><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">A</mi></mrow><mrow is="true"><mi is="true">S</mi><mo is="true">,</mo><mi is="true">S</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mi mathvariant="normal" is="true">det</mi><mo is="true">⁡</mo><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">B</mi></mrow><mrow is="true"><mi is="true">S</mi><mo is="true">,</mo><mi is="true">S</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math>, 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 2n2p−1<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mfrac is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn><mi is="true">p</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></mrow></mfrac></mrow></msup></math>-approximation to ∑Sdet⁡(AS,S)p<math><msub is="true"><mrow is="true"><mo is="true">∑</mo></mrow><mrow is="true"><mi is="true">S</mi></mrow></msub><mi mathvariant="normal" is="true">det</mi><mo is="true">⁡</mo><msup is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">A</mi></mrow><mrow is="true"><mi is="true">S</mi><mo is="true">,</mo><mi is="true">S</mi></mrow></msub><mo stretchy="false" is="true">)</mo></mrow><mrow is="true"><mi is="true">p</mi></mrow></msup></math> for any fractional number p>1<math><mi is="true">p</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">1</mn></math> in wO(wp)nO(1)<math><msup is="true"><mrow is="true"><mi is="true">w</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">w</mi><mi is="true">p</mi><mo stretchy="false" is="true">)</mo></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math> time.•We can find a 2n<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><msqrt is="true"><mrow is="true"><mi is="true">n</mi></mrow></msqrt></mrow></msup></math>-approximation to unconstrained maximum a posteriori inference in wO(wn)nO(1)<math><msup is="true"><mrow is="true"><mi is="true">w</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">w</mi><msqrt is="true"><mrow is="true"><mi is="true">n</mi></mrow></msqrt><mo stretchy="false" is="true">)</mo></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math> time.