A statistical and computational theory for robust and sparse Kalman smoothing

TL;DR

This paper develops a statistical and computational framework for robust and sparse Kalman smoothing using Piecewise Linear Quadratic penalties, enabling efficient handling of outliers and dynamic jumps in state estimation.

Contribution

It introduces a dual representation for PLQ penalties, establishing conditions for their probabilistic interpretation and extending efficient smoothing algorithms to non-smooth penalties.

Findings

Framework interprets PLQ penalties as negative log-likelihoods.

Maintains linear complexity in time series length.

Extends classical smoothing algorithms to robust and sparse settings.

Abstract

Kalman smoothers reconstruct the state of a dynamical system starting from noisy output samples. While the classical estimator relies on quadratic penalization of process deviations and measurement errors, extensions that exploit Piecewise Linear Quadratic (PLQ) penalties have been recently proposed in the literature. These new formulations include smoothers robust with respect to outliers in the data, and smoothers that keep better track of fast system dynamics, e.g. jumps in the state values. In addition to L2, well known examples of PLQ penalties include the L1, Huber and Vapnik losses. In this paper, we use a dual representation for PLQ penalties to build a statistical modeling framework and a computational theory for Kalman smoothing. We develop a statistical framework by establishing conditions required to interpret PLQ penalties as negative logs of true probability densities. Then, we present a computational framework, based on interior-point methods, that solves the Kalman smoothing problem with PLQ penalties and maintains the linear complexity in the size of the time series, just as in the L2 case. The framework presented extends the computational efficiency of the Mayne-Fraser and Rauch-Tung-Striebel algorithms to a much broader non-smooth setting, and includes many known robust and sparse smoothers as special cases.