Adaptive Kalman Filters

• Adaptive Kalman filters are algorithms that continuously update noise covariances and system parameters to optimize state estimation under uncertain and nonstationary conditions.
• They combine methods such as covariance matching, maximum-likelihood estimation, and deep learning to adapt filter parameters online for superior accuracy.
• These filters are widely used in navigation, robotics, and sensor fusion, providing robustness against abrupt disturbances and model uncertainties.

Adaptive Kalman filters are a class of algorithms extending the classical Kalman filter (KF) paradigm to accommodate scenarios in which the statistics of process and/or observation noise, system parameters, or even certain initial conditions are unknown or nonstationary. Through various estimation and learning strategies—ranging from covariance-matching and maximum-likelihood optimization to probabilistic modeling and deep learning—these filters calibrate or infer critical filter parameters online or recursively, ensuring both state estimation accuracy and robustness in volatile and poorly characterized environments.

1. Motivation and Theoretical Foundation

The standard KF operates under the premise that the state-space model, as well as process and measurement noise covariances ($Q$, $R$), are perfectly known and stationary. In practice, such full knowledge is rare. Unknown system parameters, environmental drift, abrupt disturbances, or even unobserved initial conditions can induce severe sub-optimality or divergence of classical filters. The adaptive Kalman filter (AKF) is therefore designed to mitigate these sources of uncertainty by embedding explicit mechanisms for identifying unknown covariances, model parameters, or initial states and feeding these estimates back into the filtering recursion (Lai et al., 16 Apr 2024Kutoyants, 17 Dec 2025Kutoyants, 2023).

Adaptivity can be justified both in terms of minimum mean-square error (MMSE) and from an estimation-theoretic or decision-theoretic standpoint (minimax risk under unknown parameters). For example, in the context of an unknown parameter $\theta$ in a linear-Gaussian model, three-step adaptive filtering procedures combining (i) preliminary estimation (typically via a method of moments or short-window MLE), (ii) recursive parameter updates (e.g., one-step MLE processes), and (iii) online plug-in to the KF, can achieve the optimal asymptotic lower bound for estimation error under mild ergodicity and regularity conditions (Kutoyants, 2023). Similar asymptotic efficiency can be realized in continuous-time models with unknown initial state using custom adaptations of the Kalman–Bucy equations (Kutoyants, 17 Dec 2025).

2. Algorithmic Taxonomy and Adaptation Mechanisms

The techniques deployed across the adaptive KF literature are diverse. Core categories include:

• Covariance adaptation: Estimating $Q_k$, $R_k$ online via innovation-based adaptation (sample statistics), covariance-matching, ML, Bayesian updating, or data-driven regression (e.g., RNN, GMM, Transformer) (Or et al., 2022Moghe et al., 2021Cohen et al., 18 Jan 2024Levy et al., 7 Mar 2025He et al., 9 Aug 2025Revach et al., 2021).
• Parameter adaptation: Tracking or estimating unknown system matrices or embedded parameters using gradient-based ML approaches, recursive optimization, or EM (1303.46221612.04777Fosso et al., 16 Dec 2025).
• Forgetting/adaptation to volatility: Inflating the process noise prior or otherwise adjusting the Riccati recursion using explicit forgetting factors (possibly state-dependent or robustified) to enhance reactivity to rapid model changes (Lai et al., 16 Apr 2024Narasimhappa, 2021).
• Disturbance/initial condition learning: Integrating disturbance learning (e.g., with concurrent GPs and backward smoothing) or online initial-state estimation into the filter loop (Lee, 2019Kutoyants, 17 Dec 2025).
• Robustification: Substituting Gaussian noise models by heavy-tailed analogues (e.g., Student's $t$) and adaptively scaling the covariance to maintain robustness under outliers and mixtures (Gong et al., 2023Shen et al., 2019).
• Numerical adaptation: Enhancing the numerical stability of joint state–parameter estimation through stable square-root or SVD-based sensitivities for gradient-based adaptive filtering (1303.46221612.04777).

These mechanisms can be organized according to what feature is being adapted, how (algorithmically), and by which objective or method.

Table: Selection of Adaptive Mechanisms

Adaptation Target	Mechanism	Example Source
Covariance ($Q$, $R$)	RNN/Transformer regression	(Or et al., 2022Levy et al., 7 Mar 2025Cohen et al., 18 Jan 2024)
Covariance ($Q$, $R$)	Riemannian trust-region ML	(Moghe et al., 2021)
Covariance ($Q$, $R$)	Covariance-matching, ML, EM	(Moghe et al., 2021Or et al., 2022)
For forgetting/adaptation	Variable forgetting factor RLS	(Lai et al., 16 Apr 2024)
System parameters	Gradient-based ML (SR/UD/SVD)	(1303.46221612.04777)
Initial state	Plug-in MLE/learning procedure	(Kutoyants, 17 Dec 2025)
Disturbance (unknown $g$)	Smoothing + online GP learning	(Lee, 2019)
Measurement model	Online GMM for heavy-tailed noise	(Gong et al., 2023)

3. Representative Algorithms and Mathematical Structure

3.1. Adaptive Covariance Estimation

Covariance adaptation is the most prevalent form, leveraging innovation statistics or externally learned information.

• Innovation-based adaptation estimates $Q_k$ or $R_k$ by matching the empirical innovation covariance with its theoretical model, sometimes using sliding windows and exponential smoothing (e.g., ROSE-Filter (Marchthaler, 2021, Narasimhappa, 2021)).
• Machine learning–aided adaptation: Sequence models (BiLSTM, ProcessNet, Set Transformers) infer noise covariances from measured or estimated system features (e.g., speed, curvature, raw sensor windows) and regress $Q_k$ online (Or et al., 2022Cohen et al., 18 Jan 2024Levy et al., 7 Mar 2025).
• Hybrid approaches combine innovation-based and ML-driven adaptation, e.g., scaling innovation statistics by a learned function (Cohen et al., 18 Jan 2024Levy et al., 7 Mar 2025).

3.2. Parameter and System Model Adaptation

Unknown or variable parameters in the system matrices (or initial state) necessitate embedded parameter estimation. Square-root (ASR), UD, or SVD-based sensitivity algorithms are used to stably propagate derivatives of KF statistics with respect to the unknown parameters, enabling robust gradient-based ML optimization under ill-conditioning (1303.46221612.04777).

An alternative route leverages reference-model adaptive control (MRAC) to define a tunable reference dynamics for the filter, eliminating the need for a true system model and relying instead on adaptive tracking of parameter errors using Lyapunov-stable laws (Fosso et al., 16 Dec 2025).

3.3. Non-Gaussian and Robust Filters

In the presence of heavy-tailed or outlier-prone noise, robustification via Student's $t$ likelihoods and online estimation of the covariance scaling parameter or mixture model (GMM) are used to avoid collapse of the Kalman gain. These methods often entail fixed-point iterations and inner-loop computations of effective scale parameters, achieving strong performance when noise characteristics depart from Gaussianity (Gong et al., 2023Shen et al., 2019).

4. Statistical Efficiency and Asymptotic Properties

A central goal of adaptive Kalman filtering is to maintain (or approach) the minimax efficient risk bounds achieved by clairvoyant filters, despite uncertainty in model parameters (Kutoyants, 2023Kutoyants, 17 Dec 2025). In finite- or small-sample cases, trade-offs ensue between reactivity to change (forgetting) and noise amplification; this is formalized in the filter’s recursion via the balance of process noise and estimation accuracy (Lai et al., 16 Apr 2024).

Rigorous results available for both discrete- and continuous-time linear–Gaussian models with unknown parameters or initial conditions show that one-step MLE–based adaptive plug-in filters can achieve the Hajek–Le Cam lower bound for mean-square filtering error, i.e., they are asymptotically minimax-optimal (Kutoyants, 2023Kutoyants, 17 Dec 2025).

5. Practical Implementation, Numerical Stability, and Application Domains

Practical instantiations of adaptive Kalman filters require attention to numerical stability, especially in real-time applications or under ill-conditioned models. Advanced ASR (square-root), UD, and SVD-based derivatives not only preserve positive-definiteness and symmetry of the propagated covariance matrices, but also increase robustness against round-off, especially critical in joint state–parameter estimation (1303.46221612.04777). Fixed-point or iterative inner loops may be necessary in robust Student’s $t$ or cubature variants (Gong et al., 2023Narasimhappa, 2021).

Application domains are broad and include navigation (INS/DVL, GNSS/IMU, autonomous vehicles), robust sensor fusion, acoustic echo cancellation, marine and aerial robotics, and non-stationary or partially observable dynamical systems.

6. Empirical Results and Performance Benchmarks

Adaptive Kalman filters consistently demonstrate improved root mean-square error (RMSE), especially in systems with process or measurement noise that is non-Gaussian, nonstationary, or exhibits abrupt changes. Learning-based adaptive filters using neural modules (e.g., BiLSTM, Transformers) have been shown to surpass classical innovation-based approaches and even closely approach model-based oracle performance across several high-dimensional benchmarks (Or et al., 2022Cohen et al., 18 Jan 2024Levy et al., 7 Mar 2025He et al., 9 Aug 2025).

A selection of key empirical results:

Filter Variant	Benchmark	Key RMSE/Accuracy Gain	Reference
BiLSTM adaptive KF	Oxford RobotCar	0.47 m RMSE (vs. classical: 0.62 m)	(Or et al., 2022)
Transformer A-KIT adaptive EKF	AUV INS/DVL	49–87% position improv. over fixed-Q EKF	(Cohen et al., 18 Jan 2024)
ProcessNet ANUKF	AUV INS/DVL	21.3% velocity RMSE reduction over UKF	(Levy et al., 7 Mar 2025)
TGKF robust adaptive filter	CV tracking	Lower RMSE in heavy-tailed/mixed Gaussian regimes	(Gong et al., 2023)

Computational cost varies: most machine learning–based adaptations are tractable in real-time on modern CPUs/GPUs, while robust covariance estimation or SVD-based derivatives impose higher per-step complexity that nevertheless remains manageable for moderate dimensions.

7. Limitations, Open Problems, and Future Directions

Despite substantial progress, several limitations remain. Adaptation typically does not guarantee global optimality, especially in non-linear, partially observed, or non-Gaussian regimes, and strong assumptions may be required for theoretical minimax properties. Online estimation can suffer from lag or excessive reactivity if adaptation hyperparameters are not tuned appropriately to system timescales (Lai et al., 16 Apr 2024).

Open methodological directions include joint adaptation of both process and measurement covariances in multi-sensor and nonlinear settings, continual/online learning protocols for neural adaptation modules, robustification against correlated or heteroskedastic noise, and extension of Riemannian optimization and other geometric methods to non-Gaussian models (Moghe et al., 2021He et al., 9 Aug 2025Fosso et al., 16 Dec 2025). The construction and analysis of adaptive filters in complex, high-dimensional, or structured environments remains an active and impactful research direction.