Efficient Variational Inference for Log-Gaussian Cox Processes via Voronoi Tessellation

Log-Gaussian Cox Processes (LGCP) are essential for modeling spatial point patterns but suffer from high computational costs and intractable likelihoods. We propose VoGCAM, an efficient Variational Voronoi Gaussian Coordinate Ascent Maximization framework for fitting LGCPs. Our approach first approximates the intractable integral in the LGCP likelihood using a flexible Voronoi tessellation, which incorporates observed points as integration nodes. By applying variational Gaussian approximation, we derive an Evidence Lower Bound (ELBO) that admits an explicit and closed-form expression. To optimize this objective, we develop a novel coordinate ascent algorithm that updates parameter blocks via Newton and fixed-point methods. We further enhance scalability by adopting a Nearest Neighbor Gaussian Process (NNGP) prior and utilizing the Woodbury formula to reduce matrix inversion costs. Theoretically, we prove the existence and uniqueness of the optimal solution and establish the convergence of our algorithm. Numerical experiments on synthetic and real-world data demonstrate that VoGCAM offers superior computational efficiency and inferential accuracy over state-of-the-art methods like INLA and VIFRK.