The inverse Kalman filter

TL;DR

This paper introduces the inverse Kalman filter, a novel method that enables efficient matrix-vector multiplication for covariance matrices, significantly improving computational speed in applications like nonparametric estimation of particle interactions.

Contribution

The paper presents the inverse Kalman filter and its integration with conjugate gradient methods, providing a scalable and efficient approach for covariance matrix computations in dynamic models.

Findings

Enables exact matrix-vector multiplication with linear cost.

Accelerates matrix inversion in covariance estimation.

Demonstrates effectiveness in microscopy data applications.

Abstract

We introduce the inverse Kalman filter, 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 inverse Kalman filter 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 proposed approach through applications in nonparametric estimation of particle interaction functions, using both simulations and cell trajectories from microscopy data.