[Paper Review] Banded Matrix Operators for Gaussian Markov Models in the Automatic Differentiation Era

Nicolas Durrande, Vincent Adam+3 more|arXiv (Cornell University)|Feb 26, 2019

Bayesian Modeling and Causal Inference15 citations

TL;DR

This paper introduces a set of efficient, differentiable banded matrix operators for Gaussian Markov models, enabling scalable variational inference and Hamiltonian Monte Carlo in linear time and space complexity. By integrating these operators into automatic differentiation frameworks like PyTorch or TensorFlow, the authors achieve orders-of-magnitude speedups and memory savings over dense matrix methods, particularly for large-scale models with banded precision matrices.

ABSTRACT

Banded matrices can be used as precision matrices in several models including linear state-space models, some Gaussian processes, and Gaussian Markov random fields. The aim of the paper is to make modern inference methods (such as variational inference or gradient-based sampling) available for Gaussian models with banded precision. We show that this can efficiently be achieved by equipping an automatic differentiation framework, such as TensorFlow or PyTorch, with some linear algebra operators dedicated to banded matrices. This paper studies the algorithmic aspects of the required operators, details their reverse-mode derivatives, and show that their complexity is linear in the number of observations.

Motivation & Objective

• To enable scalable, gradient-based inference (e.g., variational inference, HMC) for Gaussian Markov models with banded precision matrices.
• To overcome the computational bottleneck of dense matrix operations in Gaussian process and state space models by exploiting banded structure.
• To provide a general-purpose framework compatible with modern automatic differentiation libraries such as PyTorch and TensorFlow.
• To derive and implement reverse-mode derivatives for banded matrix operations, enabling end-to-end training of generative models.
• To demonstrate the practical scalability and predictive performance of the framework on real-world datasets with large-scale spatiotemporal data.

Proposed method

• Design and implement specialized linear algebra operators for banded matrices, including matrix-vector products, Cholesky factorization, and triangular solves.
• Derive exact reverse-mode derivatives for all core banded matrix operations to support automatic differentiation.
• Use the Cuthill-McKee algorithm to reorder nodes and minimize the bandwidth of the precision matrix, improving computational efficiency.
• Integrate the banded operators into the GPflow framework to enable end-to-end training of models with banded precisions.
• Leverage the banded structure to reduce both time complexity from O(N³) to O(Nl²) and space complexity from O(N²) to O(Nl), where l is the bandwidth.
• Apply the framework to state space models and Gaussian Markov random fields, demonstrating its generality across model classes.

Experimental results

Research questions

• RQ1 Can banded matrix operations be efficiently differentiated and integrated into automatic differentiation frameworks for scalable inference in Gaussian Markov models?
• RQ2 How does the proposed framework compare in performance and memory usage to dense matrix implementations for large-scale models?
• RQ3 Can the framework support advanced inference methods such as variational inference and Hamiltonian Monte Carlo with linear complexity?
• RQ4 What is the impact of node reordering (e.g., Cuthill-McKee) on the bandwidth and computational efficiency of banded precision matrices?
• RQ5 How well does the framework perform on real-world datasets with complex spatiotemporal dependencies?

Key findings

• The proposed framework reduces computational complexity from O(N³) to O(Nl²) and memory usage from O(N²) to O(Nl), achieving a speedup factor of approximately 10⁴ on a dataset with N=11,284 and l=117.
• The framework successfully supports Hamiltonian Monte Carlo and variational inference for banded precision matrices with linear time and space complexity.
• The variational inference model achieved a log-likelihood of -15,778.5 on the test set, outperforming both the baseline (log-likelihood: -17,146.6) and HMC (-15,873.6).
• The model effectively captures smooth latent trends and identifies high-demand urban locations (e.g., hospitals, hotels) with elevated predicted taxi demand.
• The framework enables end-to-end training of generative models using automatic differentiation, demonstrating the feasibility of integrating banded matrix operators into modern deep learning pipelines.
• The authors identify a gap in the literature for differentiating sparse inverse subsets, suggesting a need for further research in this area.