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Numerically robust square root implementations of statistical linear regression filters and smoothers

Filip Tronarp

TL;DR

The paper develops numerically robust square-root formulations for statistical linear regression filters and smoothers by replacing Cholesky downdates with QR-based updates. It presents a complete square-root SRL framework, including a cubature-based moment-matching method and an extension to state-dependent noise, with a downdate-free computation of the residual covariance factor $\bar{\Omega}$ via a single QR decomposition. The approach is validated on an ill-conditioned tracking task, where it remains robust in single precision and matches double-precision accuracy, while a downdate-based baseline fails under low precision. This work enhances reliability of Gaussian state estimation in challenging numerical regimes and broadens practical applicability of square-root filtering and smoothing techniques.

Abstract

In this article, square-root formulations of the statistical linear regression filter and smoother are developed. Crucially, the method uses QR decompositions rather than Cholesky downdates. This makes the method inherently more numerically robust than the downdate based methods, which may fail in the face of rounding errors. This increased robustness is demonstrated in an ill-conditioned problem, where it is compared against a reference implementation in both double and single precision arithmetic. The new implementation is found to be more robust, when implemented in lower precision arithmetic as compared to the alternative.

Numerically robust square root implementations of statistical linear regression filters and smoothers

TL;DR

The paper develops numerically robust square-root formulations for statistical linear regression filters and smoothers by replacing Cholesky downdates with QR-based updates. It presents a complete square-root SRL framework, including a cubature-based moment-matching method and an extension to state-dependent noise, with a downdate-free computation of the residual covariance factor via a single QR decomposition. The approach is validated on an ill-conditioned tracking task, where it remains robust in single precision and matches double-precision accuracy, while a downdate-based baseline fails under low precision. This work enhances reliability of Gaussian state estimation in challenging numerical regimes and broadens practical applicability of square-root filtering and smoothing techniques.

Abstract

In this article, square-root formulations of the statistical linear regression filter and smoother are developed. Crucially, the method uses QR decompositions rather than Cholesky downdates. This makes the method inherently more numerically robust than the downdate based methods, which may fail in the face of rounding errors. This increased robustness is demonstrated in an ill-conditioned problem, where it is compared against a reference implementation in both double and single precision arithmetic. The new implementation is found to be more robust, when implemented in lower precision arithmetic as compared to the alternative.
Paper Structure (11 sections, 3 theorems, 39 equations, 1 figure)

This paper contains 11 sections, 3 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contribution
  3. Notation
  4. Gaussian state estimation in square-root form
  5. Statistical linear regression in square root form
  6. Statistical linear regression
  7. Cubature implementation of statistical linear regression
  8. Square root implementation of statistical linear regression
  9. Extension to state-dependent noise
  10. Experimental results
  11. Conclusion

Key Result

Lemma 1

The model following model is equivalent to where the parameters $P, \Gamma$, and $\Sigma$ are given by

Figures (1)

  • Figure 1: The errors in position (top), velocity (middle), and turn rate (bottom), for the reference implementation Yaghoobi2022 (ref) and the proposed implementation (prop). The average errors across Monte Carlo trials is shown in solid, whereas the error for each trial is shown in transparent color. The floating point precision used by the algorithm is indicated in the legend by 32 and 64 for single double precision, respectively. Note that the results of the algorithms are completely indistinguishable. Except for ref32 which can not run at all because of failing Cholesky downdates, hence its absence in the figure.

Theorems & Definitions (4)

  • Lemma 1
  • Proposition 1
  • proof
  • Corollary 1