\textit{Less But Better}\\ Towards better \textit{AQ} Monitoring by learning \\ Inducing Points for Multi-Task Gaussian Processes

Air pollution is a pressing global issue affecting both human health and environmental sustainability. The high financial burden of conventional Air Quality (AQ) monitoring stations and their sparse spatial distribution necessitate advanced inferencing techniques for effective regulation and public health policies. We introduce a comprehensive framework employing Variational Multi-Output Gaussian Processes (VMOGP) with a Spectral Mixture (SM) kernel designed to model and predict multiple AQ indicators, particularly $PM_{2.5}$ and Carbon Monoxide ($CO$). Our method unifies the strengths of Multi-Output Gaussian Processes (MOGPs) and Variational Multi-Task Gaussian Processes (VMTGP) to capture intricate spatio-temporal correlations among air pollutants, thus delivering enhanced robustness and accuracy over Single-Output Gaussian Processes (SOGPs) and state-of-the-art neural attention-based methods. Importantly, by analyzing the variational distribution of auxiliary inducing points, we identify high-information geographical locales for optimized AQ monitoring frameworks. Through extensive empirical evaluations, we demonstrate superior performance in both accuracy and uncertainty quantification. Our methodology promises significant implications for urban planning, adaptive station placement, and public health policy formulation.