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Schur-MI: Fast Mutual Information for Robotic Information Gathering

Kalvik Jakkala, Jason O'Kane, Srinivas Akella

TL;DR

This paper tackles the computational bottleneck of mutual information (MI) in Gaussian-process–based robotic information gathering (RIG) for sensor placement and informative path planning. It introduces Schur-MI, a Schur-complement–based MI formulation augmented with precomputation and a noisy surrogate to avoid degeneracy, achieving per-evaluation cost $O(|\mathcal{A}|^3)$. Empirical results show up to 12.7× speedups on real bathymetry datasets and successful field trials with an autonomous surface vehicle, while preserving MI quality. By enabling online MI-based planning, Schur-MI bridges the gap between information-theoretic objectives and real-time robotic exploration, including effective handling of non-stationary kernels.

Abstract

Mutual information (MI) is a principled and widely used objective for robotic information gathering (RIG), providing strong theoretical guarantees for sensor placement (SP) and informative path planning (IPP). However, its high computational cost, dominated by repeated log-determinant evaluations, has limited its use in real-time planning. This letter presents Schur-MI, a Gaussian process (GP) MI formulation that (i) leverages the iterative structure of RIG to precompute and reuse expensive intermediate quantities across planning steps, and (ii) uses a Schur-complement factorization to avoid large determinant computations. Together, these methods reduce the per-evaluation cost of MI from $\mathcal{O}(|\mathcal{V}|^3)$ to $\mathcal{O}(|\mathcal{A}|^3)$, where $\mathcal{V}$ and $\mathcal{A}$ denote the candidate and selected sensing locations, respectively. Experiments on real-world bathymetry datasets show that Schur-MI achieves up to a $12.7\times$ speedup over the standard MI formulation. Field trials with an autonomous surface vehicle (ASV) performing adaptive IPP further validate its practicality. By making MI computation tractable for online planning, Schur-MI helps bridge the gap between information-theoretic objectives and real-time robotic exploration.

Schur-MI: Fast Mutual Information for Robotic Information Gathering

TL;DR

This paper tackles the computational bottleneck of mutual information (MI) in Gaussian-process–based robotic information gathering (RIG) for sensor placement and informative path planning. It introduces Schur-MI, a Schur-complement–based MI formulation augmented with precomputation and a noisy surrogate to avoid degeneracy, achieving per-evaluation cost . Empirical results show up to 12.7× speedups on real bathymetry datasets and successful field trials with an autonomous surface vehicle, while preserving MI quality. By enabling online MI-based planning, Schur-MI bridges the gap between information-theoretic objectives and real-time robotic exploration, including effective handling of non-stationary kernels.

Abstract

Mutual information (MI) is a principled and widely used objective for robotic information gathering (RIG), providing strong theoretical guarantees for sensor placement (SP) and informative path planning (IPP). However, its high computational cost, dominated by repeated log-determinant evaluations, has limited its use in real-time planning. This letter presents Schur-MI, a Gaussian process (GP) MI formulation that (i) leverages the iterative structure of RIG to precompute and reuse expensive intermediate quantities across planning steps, and (ii) uses a Schur-complement factorization to avoid large determinant computations. Together, these methods reduce the per-evaluation cost of MI from to , where and denote the candidate and selected sensing locations, respectively. Experiments on real-world bathymetry datasets show that Schur-MI achieves up to a speedup over the standard MI formulation. Field trials with an autonomous surface vehicle (ASV) performing adaptive IPP further validate its practicality. By making MI computation tractable for online planning, Schur-MI helps bridge the gap between information-theoretic objectives and real-time robotic exploration.
Paper Structure (19 sections, 16 equations, 9 figures, 1 algorithm)

This paper contains 19 sections, 16 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Information-Theoretic Objectives for RIG
  4. Optimization and Planning Algorithms
  5. Scalability and Efficiency in RIG
  6. Preliminaries: Gaussian Processes
  7. Problem Statement
  8. Method
  9. Standard Mutual Information Criterion
  10. Precomputation and an Alternate MI-RIG Objective
  11. Nondegenerate Schur-Complement-Based MI: Schur-MI
  12. Reference Algorithm: Greedy Maximization of Schur-MI
  13. Experiments
  14. MI-Based Methods in Discrete Environments
  15. Schur-MI with Precomputation
  16. ...and 4 more sections

Figures (9)

  • Figure 1: An autonomous surface vehicle mapping a lake during our field trials.
  • Figure 2: SMSE for sensor placement, comparing MI with Bayesian optimal design objectives. Curves show mean $\pm$ standard deviation of SMSE versus the number of waypoints; lower is better. Because the greedy methods are deterministic, results variability was negligible. These results highlight the performance advantage of MI-based objectives for discrete-space SP.
  • Figure 3: Runtime for sensor placement, comparing MI with Bayesian optimal design objectives. Curves show mean $\pm$ standard deviation of Runtime versus the number of waypoints; lower is better.
  • Figure 4: Runtime for SP across MI formulations. Curves show mean $\pm$ standard deviation of runtime versus the number of waypoints; lower is better. Schur-MI with PC is consistently the fastest MI formulation.
  • Figure 5: Runtime for SP across MI formulations. Curves show mean $\pm$ standard deviation of runtime versus the number of waypoints; lower is better. The results show that all four MI formulations are equivalent.
  • ...and 4 more figures