Algorithms and hardware for solving quadratic unconstrained binary optimization (QUBO) problems have made significant recent progress. This advancement has focused attention on formulating combinatorial optimization problems as quadratic polynomials. To improve the performance of solving large QUBO problems, it is essential to minimize the number of binary variables used in the objective function. In this paper, we propose a QUBO formulation that offers a bit capacity advantage over conventional quadratization techniques. As a key application, this formulation significantly reduces the number of binary variables required for score-based Bayesian network structure learning. Experimental results on 16 instances, ranging from 37 to 223 variables, demonstrate that our approach requires fewer binary variables than quadratization by orders of magnitude. Moreover, an annealing machine that implement our formulation have outperformed existing algorithms in score maximization.

Gaussian process regression is a powerful Bayesian nonlinear regression method. Recent research has enabled the capture of many types of observations using non-Gaussian likelihoods. To deal with various tasks in spatial modeling, we benefit from this development. Difficulties still arise when we can only access summarized data consisting of representative features, summary statistics, and data point counts. Such situations frequently occur primarily due to concerns about confidentiality and management costs associated with spatial data. This study tackles learning and inference using only summarized data within the framework of Gaussian process regression. To address this challenge, we analyze the approximation errors in the marginal likelihood and posterior distribution that arise from utilizing representative features. We also introduce the concept of sample quasi-likelihood, which facilitates learning and inference using only summarized data. Non-Gaussian likelihoods satisfying certain assumptions can be captured by specifying a variance function that characterizes a sample quasi-likelihood function. Theoretical and experimental results demonstrate that the approximation performance is influenced by the granularity of summarized data relative to the length scale of covariance functions. Experiments on a real-world dataset highlight the practicality of our method for spatial modeling.