# Exactly Sparse Gaussian Variational Inference with Application to Derivative-Free Batch Nonlinear State Estimation

arXiv: https://arxiv.org/abs/1911.08333 · HTML: https://mlsanity.com/paper/1911.08333/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Nov 2019.

## Built on by (most cited first, 7 of 7)

- [Amortized Reparametrization: Efficient and Scalable Variational Inference for Latent SDEs](https://mlsanity.com/paper/2312.10550/) (2023)
- [PKF: Probabilistic Data Association Kalman Filter for Multi-Object Tracking](https://mlsanity.com/paper/2411.06378/) (2024)
- [Low-rank variational Bayes correction to the Laplace method](https://mlsanity.com/paper/2111.12945/) (2021)
- [Distributed Variational Inference for Online Supervised Learning](https://mlsanity.com/paper/2309.02606/) (2023)
- [Aligning AI-driven discovery with human intuition](https://mlsanity.com/paper/2410.07397/) (2024)
- [Recursive Entropic Variational Inference for Nonlinear State-Space Models](https://mlsanity.com/paper/2511.15409/) (2025)
- [Gaussian Variational Inference with Non-Gaussian Factors for State Estimation: A UWB Localization Case Study](https://mlsanity.com/paper/2512.19855/) (2025)

## Abstract (arXiv)

We present a Gaussian Variational Inference (GVI) technique that can be applied to large-scale nonlinear batch state estimation problems. The main contribution is to show how to fit both the mean and (inverse) covariance of a Gaussian to the posterior efficiently, by exploiting factorization of the joint likelihood of the state and data, as is common in practical problems. This is different than Maximum A Posteriori (MAP) estimation, which seeks the point estimate for the state that maximizes the posterior (i.e., the mode). The proposed Exactly Sparse Gaussian Variational Inference (ESGVI) technique stores the inverse covariance matrix, which is typically very sparse (e.g., block-tridiagonal for classic state estimation). We show that the only blocks of the (dense) covariance matrix that are required during the calculations correspond to the non-zero blocks of the inverse covariance matrix, and further show how to calculate these blocks efficiently in the general GVI problem. ESGVI operates iteratively, and while we can use analytical derivatives at each iteration, Gaussian cubature can be substituted, thereby producing an efficient derivative-free batch formulation. ESGVI simplifies to precisely the Rauch-Tung-Striebel (RTS) smoother in the batch linear estimation case, but goes beyond the 'extended' RTS smoother in the nonlinear case since it finds the best-fit Gaussian (mean and covariance), not the MAP point estimate. We demonstrate the technique on controlled simulation problems and a batch nonlinear Simultaneous Localization and Mapping (SLAM) problem with an experimental dataset.