🤖 AI Summary
This paper addresses Gaussian process regression (GPR) under privacy constraints where only aggregated spatial data—such as representative features, summary statistics, and sample counts—are available, precluding access to raw point-level observations. To bridge this gap, we propose a novel GPR framework tailored to aggregated inputs. Its core innovation is the introduction of “sample pseudo-likelihood,” a theoretically grounded approximation that characterizes the error induced by aggregation and establishes a mapping between the variance function and non-Gaussian likelihoods—thereby relaxing GPR’s classical dependence on exact point data. Theoretical analysis reveals that model accuracy hinges on the ratio between aggregation granularity and the covariance length scale. Extensive experiments on real-world spatial datasets demonstrate that our method significantly improves modeling utility and inference reliability in privacy-preserving settings, while maintaining rigorous statistical foundations.
📝 Abstract
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.