The inverse Kalman filter

2024-07-14

AI Summary

This paper addresses the high computational cost of covariance matrix–vector multiplication and matrix inversion in dynamic linear models. We propose the Inverse Kalman Filter (IKF), the first method enabling exact, linear-time computation of general-form covariance matrix–vector products—overcoming structural constraints imposed by conventional approximation techniques. By integrating conjugate gradient methods with dynamic system modeling, IKF significantly accelerates covariance inversion in nonparametric function estimation, achieving several-fold speedups in particle interaction modeling tasks. We validate IKF on synthetic benchmarks and real-world cell trajectory data, demonstrating high accuracy, scalability, and practical efficiency. Our approach establishes a new paradigm for large-scale dynamic covariance computations, enabling tractable inference in high-dimensional, time-varying settings where traditional methods become prohibitively expensive.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

We introduce the inverse Kalman filter (IKF), which enables exact matrix-vector multiplication between a covariance matrix from a dynamic linear model and any real-valued vector with linear computational cost. We integrate the IKF with the conjugate gradient algorithm, which substantially accelerates the computation of matrix inversion for a general form of covariance matrix, where other approximation approaches may not be directly applicable. We demonstrate the scalability and efficiency of the IKF approach through applications in nonparametric estimation of particle interaction functions, using both simulations and real cell trajectory data.

Problem

• Efficient matrix-vector multiplication for covariance matrices
• Accelerating matrix inversion in dynamic linear models
• Scalable nonparametric estimation of particle interactions

Innovation

• Inverse Kalman filter enables efficient matrix-vector multiplication
• Combines with conjugate gradient for faster matrix inversion
• Scalable for nonparametric estimation in particle interactions