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On the (In)tractability of Computing Normalizing Constants for the Product of Determinantal Point Processes
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.
Computational Complexity of Normalizing Constants for the Product of Determinantal Point Processes
(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.
Some Inapproximability Results of MAP Inference and Exponentiated Determinantal Point Processes
As a corollary of the first result, we demonstrate that the normalizing constant for E-DPPs of any (fixed) constant exponent $p \geq \beta^{-1} = 10^{10^{13}}$ is $\textsf{NP}$-hard to approximate within a factor of $2^{\beta pn}$, which is in contrast to the case of $p \leq 1$ admitting a fully polynomial-time randomized approximation scheme.
Spanning Tree Constrained Determinantal Point Processes are Hard to (Approximately) Evaluate
We consider determinantal point processes (DPPs) constrained by spanning trees.
Predictive Optimization with Zero-Shot Domain Adaptation
The task is regarded as predictive optimization, but existing predictive optimization methods have not been extended to handling multiple domains.
Monotone k-Submodular Function Maximization with Size Constraints
A $k$-submodular function is a generalization of a submodular function, where the input consists of $k$ disjoint subsets, instead of a single subset, of the domain. Many machine learning problems, including influence maximization with $k$ kinds of topics and sensor placement with $k$ kinds of sensors, can be naturally modeled as the problem of maximizing monotone $k$-submodular functions. In this paper, we give constant-factor approximation algorithms for maximizing monotone $k$-submodular functions subject to several size constraints. The running time of our algorithms are almost linear in the domain size. We experimentally demonstrate that our algorithms outperform baseline algorithms in terms of the solution quality.
