Kalman Filter Is All You Need: Optimization Works When Noise Estimation Fails

1 INTRODUCTION . The Kalman Filter ( KF ) ( Kalman , 1960 ) is a celebrated method for linear filtering and prediction , with applications in many fields including tracking , guidance , navigation and control ( Zarchan and Musoff , 2000 ; Kirubarajan , 2002 ) . Due to its simplicity and robustness , it remains highly popular – with over 10,000 citations in the last 5 years alone ( Google Scholar , 2021 ) – despite the rise of many non-linear sequential prediction models ( e.g. , recurrent neural networks ) . The KF relies on the following model for a dynamic system : Xt+1 = FtXt + ωt ( ωt ∼ N ( 0 , Q ) ) Zt = HtXt + νt ( νt ∼ N ( 0 , R ) ) ( 1 ) where Xt is the state of the system at time t ( whose estimation is usually the goal ) , and its dynamics are modeled by the linear operator Ft up to the random noise ωt with covariance Q ; and Zt is the observation , which is modeled by the linear operator Ht up to the noise νt with covariance R. When the operators Ft , Ht are not assumed to depend on time , the notation may be simplified to F , H . To use the KF , one must determine the noise parameters Q , R . The filtering errors ( i.e. , estimation errors of the states { Xt } ) are minimized when Q and R correspond to the true covariance matrices of the noise ( Humpherys et al. , 2012 ) . Thus , these parameters are usually determined by noise estimation . In absence of system state data { xt } ( the ” ground truth ” ) , many methods have been suggested to determine Q and R from observations data { zt } alone ( Abbeel et al. , 2005 ; Odelson et al. , 2006 ; Zanni et al. , 2017 ; Park et al. , 2019 ) . When ground-truth data is available , however , noise estimation trivially reduces to calculation of the sample covariance matrices ( Lacey , 1998 ) : R̂ : = Cov ( { zt −Htxt } t ) , Q̂ : = Cov ( { xt+1 − Ftxt } t ) . ( 2 ) Indeed , as stated by Odelson et al . ( 2006 ) , “ the more systematic and preferable approach to determine the filter gain is to estimate the covariances from data ” . Our work focuses on such problems with ground-truth available for learning ( but not for inference after the learning , of course ) , which was motivated by a real-world Doppler radar estimation problem . Noise estimation is often not optimal : The equivalence between noise estimation and errors minimization can be proved under the standard KF assumptions – that is , known and linear dynamics and observation models ( Ft , Ht ) , with i.i.d and normally-distributed noises ( { ωt } , { νt } ) ( Humpherys et al. , 2012 ) . However , as put by Thomson ( 1994 ) , “ experience with real-world data soon convinces one that stationarity and Gaussianity are fairy tales invented for the amusement of undergraduates ” – and linearity and independence can be safely added to this list . Therefore , under realistic assumptions , the covariance of the noise does not necessarily correspond to optimal filtering . We introduce a case study in the context of radar tracking , where we demonstrate that even using the true covariance of the noise ( ” oracle ” noise-estimation ) is sub-optimal in a variety of scenarios – including very simple scenarios with relatively minor violation of the KF assumptions . In Appendices E and F , we also analyze this phenomenon analytically for two private cases ( non-linearity in Doppler radar and non-i.i.d noise in lidar ) , where the violation of a single KF assumption is shown to modify the effective noise . By providing this extensive evidence that noise-estimation is a sub-optimal way to tune a KF even in presence of system-states ground-truth data , we re-open a problem that was considered solved for decades ( Kalman , 1960 ) . We also show that seemingly small changes in the properties of the scenario may lead to major changes in the desired design of the KF , e.g. , whether to use a KF or an Extended KF ( Sorenson , 1985 ) . In certain cases , the design choices are easy to overlook ( e.g. , Cartesian vs. spherical coordinates ) , and are not trivial to make even if noticed . As a result , it is impractical to manually choose or develop a variant of the KF for every problem . Rather , we should assume that our model is sub-optimal , and leverage data to deal with the sub-optimality as robustly as possible . Optimization is optimal : We consider Q and R as model parameters that should be optimized with respect to the filtering errors ( i.e. , system-state estimation errors ) – rather than estimating the noise . While both noise estimation and errors optimization rely on exploitation of data , only the latter explicitly addresses the actual goal of solving the filtering problem . Gradient-based optimization methods are usually effective in the field of machine learning , but applying them naively to the entries of Q and R may violate the symmetry and positive-definiteness ( SPD ) constraints of the covariance matrices . Indeed , even works that come as far as optimizing Q and R ( instead of estimating the noise ) usually apply limited optimization methods , e.g. , gridsearch ( Coskun et al. , 2017 ) or diagonal restriction of the covariance matrices ( Formentin and Bittanti , 2014 ; Li et al. , 2019 ) . To address this issue , we use a parameterization based on Cholesky decomposition ( Horn and Johnson , 1985 ) , which allows us to apply gradient-based optimization to SPD matrices . This method is computationally efficient compared to other general gradient-based methods for SPD optimization ( Tsuda et al. , 2005 ; Tibshirani , 2015 ) . We demonstrate that the optimization reduces the errors of the KF consistently : over different variants of the KF , over different violations of KF assumptions , over different domains ( tracking from radar , video or lidar ) , over small and large training datasets , and even under distributional shifts between train and test datasets . Furthermore , we show that optimization improves the robustness to design decisions , by shrinking the performance gaps between different variants of the KF . As explained above , we extensively justify the need to optimize the KF in many practical problems , and suggest a simple solution which is effective , robust , computationally efficient , and relies on standard tools in supervised machine learning . As a result , we believe that in the scope of filtering problems with available ground-truth , whenever the KF assumptions are not strictly-guaranteed , the suggested optimization method should become the new standard procedure for the KF tuning . Unfair comparison : Many learning algorithms have been suggested to address non-linearity in filtering problems , e.g. , based on Recurrent Neural Networks ( RNN ) . Such works often use a linear tool such as the KF as a baseline for comparison – with tuning parameters being sometimes ignored ( Gao et al. , 2019 ) , sometimes based on noise estimation ( fa Dai et al. , 2020 ) , and sometimes optimized in a limited manner using trial-and-error ( Jamil et al. , 2020 ) or grid-search ( Coskun et al. , 2017 ) . Our findings imply that such a methodology yields over-optimistic conclusions , since the baseline is not optimized to the same level as the learning model . This may result in adoption of over-complicated algorithms with no actual added value . Instead , any learning algorithm should be compared to a baseline that is optimized using a similar method ( e.g. , gradient-descent with respect to the errors ) . Indeed , we consider an extension of the KF based on LSTM , which is the key component in many SOTA algorithms for non-linear sequential prediction in recent years ( Neu et al. , 2021 ) . For radar tracking with non-linear motion , we demonstrate how the LSTM seems to provide a significant improvement over the KF . Then , we show that the whole improvement comes from optimization of parameters , and not from the expressive non-linear architecture . In particular , this result demonstrates the competitiveness of our suggested method versus SOTA sequential prediction models . Recent works in the area of machine learning have already shown that advanced algorithms often obtain most of their improvement from implementation nuances ( Engstrom et al. , 2019 ; Andrychowicz et al. , 2020 ; Henderson et al. , 2017 ) . Our work continues this line of thinking and raises awareness to this issue in the domain of filtering problems . Contribution : We show ( empirically and analytically ) that the gold-standard noise estimation method to tune the KF given ground-truth data is often sub-optimal ; demonstrate how this leads to underevaluation of the KF compared to optimized filtering models such as neural networks ; suggest a simple method to optimize the KF , using gradient-based optimization with Cholesky parameterization ; and extensively demonstrate its improved accuracy and its robustness to model misspecification . Limitations : This work relies on the availability of ground-truth system-states in the training data to allow supervised learning . Ground-truth data is often available from simulations , controlled experiments or manual labeling . Within this scope , we show that although noise estimation is straightforward ( Equation 2 ) , it is often not the right task to address . Another limitation is in the theoretical guarantees of gradient-based optimization ( see Appendix L ) , despite its wide success in many fields . The paper is organized as follows : Section 2 reviews the KF . Section 3 introduces our method for efficient KF optimization . Section 4 justifies the necessity of KF optimization through a detailed case study . Section 5 presents a neural version of the KF which reduces the errors compared to a standard KF – but not when compared to an optimized KF . Section 6 discusses related works . 2 PRELIMINARIES : THE KALMAN FILTER ALGORITHM . The KF algorithm ( Kalman , 1960 ; Humpherys et al. , 2012 ) relies on Equation 1 for a dynamic system model . It keeps an estimate of the state Xt , represented as the mean xt and covariance Pt of a normal distribution . As shown in Figure 1 , it alternately predicts the next state using the dynamics model ( prediction step ) , and processes new information from incoming observations ( update or filtering step ) . The KF yields optimal state estimations – but only under a restrictive set of assumptions ( Kalman , 1960 ) , as specified in Def- inition 2.1 . Note that normality of the noise is excluded since it is not necessary for optimality ( Humpherys et al. , 2012 ) , although it is also often assumed . Definition 2.1 ( KF assumptions ) . Ft , Ht of Equation 1 are known matrices that do not depend on the state ( i.e. , correspond to linear models ) ; ωt , νt are i.i.d random variables with zero-mean and constant , known covariance matrices Q , R , respectively ; and the initial state distribution is known . Certain assumptions violating in Definition 2.1 can be partially handled by variations of the KF , such as the Extended KF ( EKF ) ( Sorenson , 1985 ) which replaces the linear models Ft , Ht with local linear approximations , and the Unscented KF ( UKF ) ( Wan and Van Der Merwe , 2000 ) which applies the filtering through sigma-points sampled from the estimated distribution . The use of multiple tracking models alternately is also possible using switching mechanisms ( Mazor et al. , 1998 ) . While Ft and Ht are usually determined based on domain knowledge , Q and R are often estimated from data as the covariance of the noise . As mentioned in Section 1 , this can be done using Equation 2 if ground-truth data is available , or using more sophisticated methods otherwise ( Odelson et al. , 2006 ; Feng et al. , 2014 ; Park et al. , 2019 ) . See Appendix A for a detailed introduction of the KF and recurrent neural networks ( RNN , LSTM ) .

Summary: This paper targets the design of a classical filtering method - the Kalman filter (KF). The linearity assumption is a strong limitation of KF models although a wide range of variants have been demonstrated for non-linear systems. Different from these studies, the authors focused on the estimation of the noise models in the KF(-class) models, in order to improve the accuracy and robustness of the KF estimates, through an optimization method. The proposed approach was assessed on a benchmark dataset, and the results demonstrated the superiority of the proposed method. Also theoretical analyses are provided in the appendices.