Shedding Light on Factor Graph Optimization: An Analysis of the Structure and Dimensionality of the Optimization Problem for GNSS-PPP

This paper investigates the application of Factor Graph Optimization (FGO) to Precise Point Positioning (PPP), with a focus on the structural and dimensional characteristics of the resulting optimization problem. We begin by outlining the theoretical foundation of FGO, framing the Maximum A Posteriori estimation as a factorized representation of probability densities that reduces to a least-squares problem under Gaussian noise assumptions. We discuss the importance of exploiting sparsity for the efficient solution via QR or Cholesky factorization, and highlight the connection between the structure of the factor graph and the Jacobian matrix. A simplified single-frequency PPP model is introduced, and we compare different carrier phase ambiguity parametrization (per-track and per-epoch) by analyzing their corresponding factor graphs, Jacobians, normal equation matrices, and Cholesky factors. Our findings emphasize the critical role of variable ordering in maintaining sparsity, particularly when estimating per-track ambiguities. We show that placing ambiguity variables last yields a sparse Cholesky factor, whereas placing them first leads to a dense Cholesky factor. In contrast, per-epoch ambiguity estimation naturally supports sparse solutions. We further explore how the matrix-graph duality enables partial QR factorization for GNSS-PPP, revealing opportunities for parallel computation. Finally, we address incremental updates to the factor graph, demonstrating how new observations can be incorporated without compromising sparsity. These insights support designing custom solvers for specific problems or a more thoughtful and effective use of existing FGO libraries.