KILO-EKF: Koopman-Inspired Observation Filtering

• The paper introduces KILO-EKF, which leverages Koopman lifting to transform nonlinear observation models into a linear-Gaussian framework, enabling robust state estimation.
• It details a closed-form MAP learning method for the observation mapping, ensuring scalability and real-time performance without iterative optimization.
• Experiments on quadrotor localization demonstrate a 10–20% RMSE reduction and improved covariance consistency compared to traditional EKF baselines.

The Koopman-Inspired Learned Observations Extended Kalman Filter (KILO-EKF) is an algorithmic framework for nonlinear state estimation that integrates Koopman-theoretic lifting of observation models with the established recursive structure of the Extended Kalman Filter (EKF). KILO-EKF addresses the challenge of nonlinear or poorly calibrated observation processes, learning a linear-Gaussian measurement model in a high-dimensional feature space directly from data. This approach enables scalable, real-time filtering in systems exhibiting complex or unknown sensor characteristics, as frequently encountered in robotics and control applications (Guo et al., 18 Jan 2026, Li et al., 2 Dec 2025).

1. Theoretical Framework and Problem Motivation

Traditional EKF implementations necessitate analytic or hand-engineered observation models and noise covariances. In practice, this poses challenges when sensors are uncalibrated, affected by environmental perturbations, or exhibit nonlinear distortions—circumstances common with ultra-wideband (UWB), radio-frequency identification (RFID), and various vision or acoustic sensors. Analytical models become brittle or inflexible, and imperfect models can lead to divergence or reduced accuracy of the filter state estimates.

KILO-EKF leverages insights from Koopman operator theory to circumvent explicit parametric sensor modeling. The core premise is to lift the nonlinear measurement process into a feature space where the measurement becomes linear with respect to the state. This is achieved by applying a differentiable feature map $\phi$ to raw observations, yielding a linear measurement model of the form $\phi(y_k) \approx H x_k + v_k$, with $v_k$ Gaussian (Guo et al., 18 Jan 2026). The resulting observation model, once learned, is seamlessly compatible with the recursive EKF pipeline.

2. State-Space Models and EKF Formulation

KILO-EKF operates on a discrete-time nonlinear process model: $$x_{k+1} = f(x_k, u_k) + w_k, \qquad w_k \sim \mathcal N(0, Q)$$ where $x_k$ is the state, $u_k$ denotes known control inputs, and $w_k$ is process noise with known covariance. The standard EKF prediction step propagates mean and covariance through the Jacobian of $f$, yielding: $$\hat x_{k|k-1} = f(\hat x_{k-1|k-1}, u_{k-1}), \qquad P_{k|k-1} = F_{k-1} P_{k-1|k-1} F_{k-1}^T + Q$$ with $F_{k-1}$ being the state transition Jacobian.

The Koopman-inspired measurement model postulates the existence of a feature map $\phi : \mathbb{R}^m \to \mathbb{R}^d$, such that

$$\phi(y_k) = H x_k + v_k, \qquad v_k \sim \mathcal N(0, R)$$

where $H$ is a learned linear mapping and $R$ the measurement noise covariance. The EKF correction operates in this lifted observation space, exploiting the learned linear-Gaussian structure (Guo et al., 18 Jan 2026, Li et al., 2 Dec 2025).

3. Model Learning: Koopman-Inspired Lifting and Closed-Form Estimation

The central learning step in KILO-EKF requires synchronized pairs of true states and raw sensor observations, typically collected using ground-truth systems such as motion capture. The procedure is:

• Construct feature responses $Y = [\phi(y^{(1)}), \ldots, \phi(y^{(P)})]$ and state matrix $X = [x^{(1)}, \ldots, x^{(P)}]$ from $P$ training samples.
• Estimate $H$ and $R$ via regularized closed-form maximum a posteriori (MAP) solutions: $H = Y X^T (X X^T + P \tau_H I_n)^{-1}$

$$R = \frac{1}{P}(Y - H X)(Y - H X)^T + \tau_H H H^T + \tau_R I_d$$

where $\tau_H$ and $\tau_R$ are regularization hyperparameters. The computational cost is linear in the number of samples, and the formulation avoids iterative optimization or risk of local minima. Feature lifting $\phi$ is fully differentiable; choices include random Fourier features or basis expansions with polynomials or trigonometric functions (Guo et al., 18 Jan 2026).

4. EKF Update and Real-Time Inference

The update phase of KILO-EKF employs the learned $H$ and $R$ in a standard EKF correction step. The innovation Jacobian with respect to the state reduces to $H$, under the assumption that the raw observation $y_k$ is independent of the current filter state: $H_k = H$ The recursive update equations are: $$K_k = P_{k|k-1} H_k^T (H_k P_{k|k-1} H_k^T + R)^{-1}$$

$$\hat x_{k|k} = \hat x_{k|k-1} + K_k (\phi(y_k) - H_k \hat x_{k|k-1})$$

$P_{k|k} = (I - K_k H_k) P_{k|k-1}$

This update maintains real-time performance, as all computations involve straightforward matrix multiplications and inversions of fixed-size matrices. Training and inference retain the structure and computational efficiency of a standard EKF, enabling direct deployment (Guo et al., 18 Jan 2026).

5. Experimental Validation and Comparative Analysis

KILO-EKF was validated on a real-world quadrotor localization problem, utilizing body-frame IMU inputs and measurements from both UWB and downward-facing laser sensors. The method was compared against several EKF baselines:

• CAD‐EKF: employing exact CAD-based anchor/tag offsets;
• MisCAD‐EKF: with perturbed tag offsets;
• DataCal‐EKF: calibrated with batch nonlinear least squares.

Evaluation metrics included trajectory RMSE (position, orientation) and normalized estimation error squared (NEES) per filter step. KILO-EKF demonstrated consistently superior accuracy (10–20% reduction in RMSE over DataCal‐EKF) and enhanced covariance consistency (NEES histogram concentrated near 1). Ablation studies established that increasing the number of random features or utilizing more training data steadily improves filter accuracy. These results underscore the utility of data-driven Koopman lifting for observation modeling, particularly in settings where geometric sensor calibration is imperfect or infeasible (Guo et al., 18 Jan 2026).

6. Relationships to KALIKO and Related Koopman Methods

KILO-EKF is conceptually related to the Kalman-Implicit Koopman Operator (KALIKO) approach for modeling nonlinear dynamical systems via globally linear embeddings (Li et al., 2 Dec 2025). KALIKO jointly learns a finite-dimensional Koopman operator $K$ and a nonlinear decoder $h_\psi$ (often a neural network) for reconstructing measurements. Unlike KILO-EKF’s closed-form observation map, KALIKO employs a filter–smooth–predict–decode training strategy, back-propagating through the Kalman filter and smoother to optimize both $K$ and $h_\psi$ with respect to trajectory prediction losses. The EKF is used for inference in both frameworks, but KILO-EKF targets learned measurement models while maintaining an explicit nonlinear state-space evolution, whereas KALIKO enforces linearity in the latent dynamical evolution.

Both methodologies highlight the advantage of using Koopman-inspired representations for accurate, data-driven modeling of nonlinear systems in a filtering context, and demonstrate the integration of such learned representations with classical state estimators to improve robustness and scalability (Guo et al., 18 Jan 2026, Li et al., 2 Dec 2025).

7. Advantages, Limitations, and Prospective Extensions

KILO-EKF offers several strengths:

• Nonparametric, closed-form learning enables flexible modeling of complex, biased, or nonlinear sensor effects.
• Training scales linearly with data size and does not require iterative solvers.
• Retains the recursive, real-time nature of EKF for online inference.
• Outperforms traditional and data-calibrated EKF baselines under sensor model error.

Identified limitations include:

• Dependence on availability of ground-truth state labels for training.
• Performance sensitivity to the choice and design of the lifting function $\phi$.
• Assumption of approximate linearity and Gaussianity in the lifted observation space.

Proposed extensions include the incorporation of sparse feature selection (e.g., via SINDy), extension to vision or LiDAR sensors with highly nonlinear observation manifolds, and integration into simultaneous localization and mapping (SLAM) backends through learned measurement factors (Guo et al., 18 Jan 2026).

In summary, KILO-EKF represents a significant advance in bridging the gap between classical recursive filtering and modern data-driven observation modeling, by exploiting Koopman-inspired linearization of sensor effects in the observation space and delivering scalable, high-accuracy state estimation.