Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient Sensor Placement from Regression with Sparse Gaussian Processes in Continuous and Discrete Spaces

Kalvik Jakkala, Srinivas Akella

TL;DR

The paper tackles efficient sensor placement for spatiotemporally correlated fields under sparse labeling by formulating the problem as variational inference with sparse Gaussian processes. It introduces Continuous-SGP, Greedy-SGP, and Discrete-SGP, exploiting a differentiable ELBO to optimize $m$ inducing points that correspond to sensor locations, and, when needed, maps continuous solutions to a discrete candidate set via an assignment problem. Across four real datasets, the SVGP-based methods achieve reconstruction quality and mutual information on par with or better than MI-based baselines while delivering order-of-magnitude speedups, especially in large or 3D problems. The approach also provides differentiability for integration with neural networks and potential extensions to spatiotemporal, obstacle-aware, and robotic informative path planning contexts.

Abstract

The sensor placement problem is a common problem that arises when monitoring correlated phenomena, such as temperature, precipitation, and salinity. Existing approaches to this problem typically formulate it as the maximization of information metrics, such as mutual information~(MI), and use optimization methods such as greedy algorithms in discrete domains, and derivative-free optimization methods such as genetic algorithms in continuous domains. However, computing MI for sensor placement requires discretizing the environment, and its computation cost depends on the size of the discretized environment. These limitations restrict these approaches from scaling to large problems. We present a novel formulation to the SP problem based on variational approximation that can be optimized using gradient descent, allowing us to efficiently find solutions in continuous domains. We generalize our method to also handle discrete environments. Our experimental results on four real-world datasets demonstrate that our approach generates sensor placements consistently on par with or better than the prior state-of-the-art approaches in terms of both MI and reconstruction quality, all while being significantly faster. Our computationally efficient approach enables both large-scale sensor placement and fast robotic sensor placement for informative path planning algorithms.

Efficient Sensor Placement from Regression with Sparse Gaussian Processes in Continuous and Discrete Spaces

TL;DR

The paper tackles efficient sensor placement for spatiotemporally correlated fields under sparse labeling by formulating the problem as variational inference with sparse Gaussian processes. It introduces Continuous-SGP, Greedy-SGP, and Discrete-SGP, exploiting a differentiable ELBO to optimize inducing points that correspond to sensor locations, and, when needed, maps continuous solutions to a discrete candidate set via an assignment problem. Across four real datasets, the SVGP-based methods achieve reconstruction quality and mutual information on par with or better than MI-based baselines while delivering order-of-magnitude speedups, especially in large or 3D problems. The approach also provides differentiability for integration with neural networks and potential extensions to spatiotemporal, obstacle-aware, and robotic informative path planning contexts.

Abstract

The sensor placement problem is a common problem that arises when monitoring correlated phenomena, such as temperature, precipitation, and salinity. Existing approaches to this problem typically formulate it as the maximization of information metrics, such as mutual information~(MI), and use optimization methods such as greedy algorithms in discrete domains, and derivative-free optimization methods such as genetic algorithms in continuous domains. However, computing MI for sensor placement requires discretizing the environment, and its computation cost depends on the size of the discretized environment. These limitations restrict these approaches from scaling to large problems. We present a novel formulation to the SP problem based on variational approximation that can be optimized using gradient descent, allowing us to efficiently find solutions in continuous domains. We generalize our method to also handle discrete environments. Our experimental results on four real-world datasets demonstrate that our approach generates sensor placements consistently on par with or better than the prior state-of-the-art approaches in terms of both MI and reconstruction quality, all while being significantly faster. Our computationally efficient approach enables both large-scale sensor placement and fast robotic sensor placement for informative path planning algorithms.
Paper Structure (25 sections, 1 theorem, 21 equations, 12 figures, 3 algorithms)

This paper contains 25 sections, 1 theorem, 21 equations, 12 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Related Work
  4. Gaussian processes
  5. Method
  6. Theoretical Foundation
  7. Continuous-SGP: Continuous Space Solutions
  8. Greedy-SGP: Greedy Discrete Space Solutions
  9. Discrete-SGP: Gradient-based Discrete Space Solutions
  10. Experiments
  11. Conclusion
  12. Theory
  13. Preliminary
  14. Properties of Entropy
  15. Submodularity
  16. ...and 10 more sections

Key Result

Theorem 1

BurtRW20 Suppose $N$ training inputs are drawn i.i.d according to input density $p(\mathbf{x})$, and $k(\mathbf{x, x}) < v$ for all $\mathbf{x} \in \mathbf{X}$. Sample $M$ inducing points from the training data with the probability assigned to any set of size $M$ equal to the probability assigned to where $C = N \sum^\infty_{m=M+1} \lambda_m$, $\lambda_m$ are the eigenvalues of the integral operat

Figures (12)

  • Figure 1: The mean and standard deviation of the RMSE vs number of sensors for the Intel, precipitation, soil, and salinity datasets (lower is better).
  • Figure 2: The mean and standard deviation of the Runtime vs number of sensors for the Intel, precipitation, soil, and salinity datasets (lower is better).
  • Figure 3: Comparison of the MI and SVGP's lower bound (ELBO) for the soil and salinity datasets. The mean and standard deviation of the MI vs number of sensors (a), (c) and SVGP's lower bound (ELBO) vs number of sensors (b), (d).
  • Figure 4: A non-stationary environment. (a) Ground truth. Reconstructions from the Continuous-SGP solutions for (b) 32 sensing locations with a stationary RBF kernel, and (c) 9 and (d) 16 sensing locations with the neural kernel. The black pentagons represent the solution placements.
  • Figure 5: Placements for 500 sensors generated using the Continuous-SGP approach with an ST-SVGP. The red points are the sensor placements projected onto the 2D map using cylindrical equal-area projection.
  • ...and 7 more figures

Theorems & Definitions (1)

  • Theorem 1