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

• URL: http://arxiv.org/abs/2111.14148v1
• Date: Sun, 28 Nov 2021 14:08:25 GMT
• Title: Computational Complexity of Normalizing Constants for the Product of Determinantal Point Processes
• Authors: Naoto Ohsaka and Tatsuya Matsuoka
• Abstract summary: We study the computational complexity of computing the normalizing constant. We show that $sum_Sdet(bf A_S,S)p$ exactly for every (fixed) positive even integer $p$ is UP-hard and Mod$_3$P-hard.
• Score: 12.640283469603357
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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: (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.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Measuring quantum relative entropy with finite-size effect [53.64687146666141]
  We study the estimation of relative entropy $D(rho|sigma)$ when $sigma$ is known. Our estimator attains the Cram'er-Rao type bound when the dimension $d$ is fixed.
  arXiv  Detail & Related papers  (2024-06-25T06:07:20Z)
• Partially Unitary Learning [0.0]
  An optimal mapping between Hilbert spaces $IN$ of $left|psirightrangle$ and $OUT$ of $left|phirightrangle$ is presented. An iterative algorithm for finding the global maximum of this optimization problem is developed.
  arXiv  Detail & Related papers  (2024-05-16T17:13:55Z)
• Randomized and Deterministic Attention Sparsification Algorithms for Over-parameterized Feature Dimension [18.57735939471469]
  We consider the sparsification of the attention problem. For any super large feature dimension, we can reduce it down to the size nearly linear in length of sentence.
  arXiv  Detail & Related papers  (2023-04-10T05:52:38Z)
• Classical shadows of fermions with particle number symmetry [0.0]
  We provide an estimator for any $k$-RDM with $mathcalO(k2eta)$ classical complexity. Our method, in the worst-case of half-filling, still provides a factor of $4k$ advantage in sample complexity.
  arXiv  Detail & Related papers  (2022-08-18T17:11:12Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)
• Model-Free Reinforcement Learning: from Clipped Pseudo-Regret to Sample Complexity [59.34067736545355]
  Given an MDP with $S$ states, $A$ actions, the discount factor $gamma in (0,1)$, and an approximation threshold $epsilon > 0$, we provide a model-free algorithm to learn an $epsilon$-optimal policy. For small enough $epsilon$, we show an improved algorithm with sample complexity.
  arXiv  Detail & Related papers  (2020-06-06T13:34:41Z)
• Statistically Efficient, Polynomial Time Algorithms for Combinatorial Semi Bandits [3.9919781077062497]
  We consider semi-bandits over a set of arms $cal X subset 0,1d$ where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields the smallest known regret bound $R(T) = cal OBig( d (ln m)2 (ln T) over Delta_min Big)$, but it has computational complexity $cal O(|cal X|)$ which is typically exponential in $d$, and cannot
  arXiv  Detail & Related papers  (2020-02-17T21:32:04Z)
• Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm [100.11971836788437]
  We study the fixed-support Wasserstein barycenter problem (FS-WBP) We develop a provably fast textitdeterministic variant of the celebrated iterative Bregman projection (IBP) algorithm, named textscFastIBP.
  arXiv  Detail & Related papers  (2020-02-12T03:40:52Z)
• On the Complexity of Minimizing Convex Finite Sums Without Using the Indices of the Individual Functions [62.01594253618911]
  We exploit the finite noise structure of finite sums to derive a matching $O(n2)$-upper bound under the global oracle model. Following a similar approach, we propose a novel adaptation of SVRG which is both emphcompatible with oracles, and achieves complexity bounds of $tildeO(n2+nsqrtL/mu)log (1/epsilon)$ and $O(nsqrtL/epsilon)$, for $mu>0$ and $mu=0$
  arXiv  Detail & Related papers  (2020-02-09T03:39:46Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.