Kalman Evolve: Adaptive Kalman Filtering

• Kalman Evolve is a framework that retains the recursive structure of the Kalman filter while adapting the update mechanism to overcome the limitations of affine estimators.
• It utilizes a two-stage approach combining calibration of noise parameters with LLM-guided evolutionary search to discover improved, non-affine update programs.
• Empirical evaluations in Doppler radar, LiDAR, and pedestrian tracking demonstrate up to a 12% reduction in RMSE over traditional methods, validating its practical efficiency.

Kalman Evolve denotes a line of research that preserves the recursive predict–update skeleton of Kalman filtering while allowing the estimator itself to change with sensing conditions, data regime, or program structure. In its most explicit formulation, Kalman Evolve is a framework for discovering improved Kalman-style algorithms by jointly optimizing noise parameters and the update structure, motivated by the claim that realistic sensing problems such as Doppler radar and LiDAR create a performance gap that cannot be eliminated by tuning $Q$ and $R$ alone (Saketos et al., 26 May 2026). In related work, the same broader idea appears as context-conditioned gain modulation, unsupervised online adaptation, latent linearization, and other methods that make Kalman-style estimation “evolve” without abandoning recursive filtering (Ni et al., 2023, Revach et al., 2021).

1. Classical recursion and the source of the performance gap

The starting point is the standard linear state-space model

$$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$

$$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$

with Kalman recursion

$$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$

$$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$

$$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$

Kalman Evolve is built around the claim that the practical limitation of this recursion is often structural rather than merely parametric: even an Optimized Kalman Filter that learns $Q$ and $R$ remains in the same affine estimator class, because the update is still an affine transformation of the innovation (Saketos et al., 26 May 2026).

This point is emphasized in sensing settings where the measurement process is nonlinear even when the dynamics are linear or approximately linear. The cited examples are Doppler radar, where measurements depend on radial velocity, and LiDAR, where measurements are naturally in polar coordinates and become nonlinear after conversion to Cartesian coordinates. A common misconception is that such degradation can be removed by covariance tuning alone. The Kalman Evolve position is that this is generally false: if the posterior mean is nonlinear in the observation, then no choice of $Q$ and $R$0 inside a standard Kalman filter can recover the Bayes/MMSE estimator (Saketos et al., 26 May 2026).

This suggests that “evolution” in this context does not mean discarding Kalman filtering. It means retaining the recursive form while changing what is allowed inside the update.

2. Formal critique of affine updates

The central analytical contribution of the Kalman Evolve framework is a proof strategy showing that affine estimators are strictly suboptimal under common nonlinear sensing models. For the Doppler setting, the paper considers a jointly Gaussian, isotropic latent position $R$1 and velocity $R$2, with observation

$R$3

It defines the affine class

$R$4

and states that the Bayes estimator

$R$5

is not affine, hence

$R$6

The argument uses the induced nonlinear range information in the Doppler term and rotational equivariance to show that an affine equivariant estimator cannot exploit the scalar Doppler channel in the way the Bayes estimator does (Saketos et al., 26 May 2026).

For LiDAR, the same conclusion is derived in a 2D Gaussian latent model with noisy polar measurements

$R$7

converted to Cartesian observations

$R$8

The paper states that $R$9 is not Gaussian, that the Bayes estimator $$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$0 is not affine, and therefore affine estimators $$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$1 are strictly suboptimal in mean-squared error (Saketos et al., 26 May 2026).

A related next-state prediction result analyzes transition-model mismatch: $$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$2 with error recursion

$$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$3

Under stability and bounded mismatch/noise energy assumptions, the paper derives an upper bound with a transient term and a persistent mismatch-driven term. The significance is not merely technical. It motivates modifying the update structure itself to reduce the long-run residual error rather than treating mismatch as a problem of covariance calibration alone (Saketos et al., 26 May 2026).

3. Kalman Evolve as interpretable algorithm discovery

Kalman Evolve operationalizes this critique through a two-stage procedure. Stage 1 calibrates a strong baseline by estimating $$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$4 and $$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$5 using either least-squares covariance estimation or the OKF procedure. Stage 2 then performs LLM-guided evolutionary search over update programs rather than over covariance parameters alone. Candidate algorithms are stored in a database, prompts are built from the problem specification and sampled candidate algorithms, the LLM generates mutations and combinations, and the best candidates are retained (Saketos et al., 26 May 2026).

The implementation details are explicit. The search uses DeepSeek 16B, is distributed across 4 islands/GPU workers, each island runs 10 iterations of sampling/mutation/database update per cycle, weaker islands are reinitialized from the best candidate of the strongest island, and the process is repeated for 20 cycles (Saketos et al., 26 May 2026). The key methodological point is that the LLM acts as a structured prior over program space rather than as the estimator itself.

The discovered programs preserve the classical recursion but introduce non-affine residual processing and adaptive scaling. The retained semantic blocks include innovation

$$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$6

Kalman gain

$$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$7

state update

$$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$8

covariance update

$$x_t = F x_{t-1} + w_{t-1}, \quad w_{t-1} \sim \mathcal{N}(0, Q),$$9

and prediction covariance

$$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$0

The gate and scaling factors are computed from residual statistics such as means, variances, fourth moments, clipping, and $$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$1-type nonlinearities (Saketos et al., 26 May 2026).

This emphasis on symbolic, human-readable modifications links Kalman Evolve to earlier algorithm-discovery work that combined Cartesian Genetic Programming and LLM-assisted evolutionary search to rediscover the Kalman filter under its classical assumptions and to evolve interpretable alternatives when those assumptions were violated. In that earlier study, the framework converged to near-optimal solutions when Kalman optimality assumptions held, while under half-Gaussian noise the LLM-assisted method achieved MSE around $$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$2 versus the Kalman filter’s $$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$3, and under delayed observations CGP achieved about $$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$4 versus the Kalman filter’s $$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$5 (Saketos et al., 13 Aug 2025). A plausible implication is that Kalman Evolve extends this earlier program-discovery agenda from proof-of-concept symbolic recovery to benchmarked state-estimation design.

4. Benchmarks, baselines, and empirical standing

Kalman Evolve is evaluated on three main problem families: Doppler radar tracking, LiDAR-based localization/state estimation, and pedestrian tracking on MOT20, with KF, OKF, KalmanNet, LSTM, and observation-based baselines used in different settings (Saketos et al., 26 May 2026). The main metric is RMSE, and the framework is reported to achieve up to $$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$6 reduction in RMSE over strong baselines.

In Doppler radar, five benchmarks of increasing difficulty—Toy, Close, Const_v, Const_a, and Free—were used. For state estimation, the reported RMSEs were: Toy $$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$7 for KF, $$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$8 for OKF, and $$z_t = H x_t + v_t, \quad v_t \sim \mathcal{N}(0, R),$$9 for KE; Close $$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$0, $$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$1, and $$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$2; Const_v $$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$3, $$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$4, and $$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$5; Const_a $$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$6, $$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$7, and $$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$8; Free $$\hat{x}_{t+1}^P := \tilde{F}_t \hat{x}_t,\qquad \hat{P}_{t+1}^P := \tilde{F}_t \hat{P}_t \tilde{F}_t^{T} + \hat{Q},$$9, $$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$0, and $$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$1. The reported improvements of KE over OKF ranged from about $$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$2 on Toy to about $$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$3 on Free. For next-state prediction, gains generally increased as motion complexity and mismatch increased, although Toy NSP was a case where KE did not beat OKF (Saketos et al., 26 May 2026).

In synthetic LiDAR-like 2D tracking, the RMSE table gave state-estimation errors of $$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$4 for KF, $$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$5 for OKF, $$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$6 for KE, $$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$7 for OKE, $$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$8 for KalmanNet, $$K := \hat{P}_t^P \tilde{H}_t^{T} (\tilde{H}_t \hat{P}_t^P \tilde{H}_t^{T} + \hat{R})^{-1},$$9 for LSTM, and $$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$0 for raw observations. For next-state prediction, the corresponding values were $$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$1, $$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$2, $$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$3, $$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$4, $$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$5, $$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$6, and $$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$7. On the NCLT dataset, state-estimation RMSE was $$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$8 for KF, $$\hat{x}_t := \hat{x}_t^P + K(z_t - \tilde{H}_t \hat{x}_t^P), \qquad \hat{P}_t := (I - K\tilde{H}_t)\hat{P}_t^P.$$9 for OKF, $Q$0 for KE, and $Q$1 for OKE; next-state prediction was $Q$2, $Q$3, $Q$4, and $Q$5. Runtime on LiDAR was reported as around 2.5–3 seconds for KF, OKF, KE, and OKE, while LSTM and KalmanNet were much slower (Saketos et al., 26 May 2026).

On MOT20, used for next-state prediction only, the reported RMSEs were $Q$6 for KF, $Q$7 for OKF, $Q$8 for KE, $Q$9 for OKE, $R$0 for LSTM, and $R$1 for KalmanNet. The paper reports about 8% reduction relative to OKF and a $R$2-test on per-trajectory squared error differences with $R$3-values $R$4 (Saketos et al., 26 May 2026).

The empirical position of Kalman Evolve is therefore not that of a universal replacement for all learned or model-based filters. It is a structured alternative for cases where the KF recursion is computationally attractive but the affine update family is too restrictive.

5. Adaptive learned Kalman filters as a parallel “evolution” line

A distinct but closely related line of work makes Kalman-style estimators evolve through learned context adaptation rather than symbolic program search. Adaptive KalmanNet conditions the learned gain computation on the strength of the weak noise ratio

$R$5

and uses a compact hypernetwork to generate context-dependent modulation weights

$R$6

so that a generic layer

$R$7

becomes

$R$8

The resulting gain network is $R$9, learned with a two-stage procedure in which the KalmanNet backbone is first trained on a pseudo-stationary subset and the hypernetwork is then trained over varying noise regimes. Numerical evaluation shows consistent state-estimation performance across a continuous range of noise distributions even when training uses only limited noise settings (Ni et al., 2023).

Unsupervised Learned Kalman Filtering pushes the same adaptive theme in another direction: it keeps the KalmanNet architecture but removes the need for ground-truth state labels. Instead of a state loss, it minimizes the observation-prediction error

$Q$0

using the internally computed innovation $Q$1. The reported findings are that unsupervised KalmanNet reaches the MMSE lower bound in linear experiments, approaches the KF with full domain knowledge, is within about 0.5 dB of the extended KF on the nonlinear Lorenz attractor example, and can adapt a pre-trained model online every $Q$2 samples to a changed noise regime without additional labels (Revach et al., 2021).

Adaptive Multi-modal KalmanNet extends these ideas to autonomous driving. AM-KNet introduces sensor-specific measurement modules for radar, lidar, and camera; a 27-dimensional context vector encoding target type, motion state, and relative pose; and a covariance branch based on the Joseph form

$Q$3

It is trained with component-wise state error, physical weighting terms, and negative log-likelihood losses on both state residuals and innovations. On View-of-Delft, AM-KNet+CM achieved position MAE $Q$4 m and $Q$5 m and position NEES consistency 76.97%; on nuScenes it achieved position MAE $Q$6 m and $Q$7 m, velocity MAE $Q$8 m/s and $Q$9 m/s, and position NEES consistency 60.27% (Mehrfard et al., 2 Apr 2026).

Taken together, these works show that not all “evolving” Kalman methods pursue the same mechanism. Some evolve gain computation through hypernetwork modulation, some evolve via unsupervised online updating, and some evolve via sensor- and context-aware architecture design. What they share is preservation of the KF flow together with an explicit adaptation law.

6. Broader generalizations and the evolving Kalman paradigm

Beyond symbolic search and adaptive learned gains, several other research directions extend the same basic idea: retain Kalman-style recursion, but change the representation, geometry, or surrounding dynamical system so that the recursion becomes appropriate again. The Diffusion Maps Kalman Filter learns virtual coordinates from data using diffusion maps, in which a class of gradient-flow systems with isotropic diffusion becomes approximately linear; a Kalman filter is then run in those coordinates, yielding non-parametric state estimation that outperformed competing non-parametric methods in the reported tracking problems and produced meaningful representations of rat position from hippocampal neural activity (Shnitzer et al., 2017). LAE-EnKF learns a nonlinear encoder–decoder, a stable linear latent evolution operator, and a consistent latent observation mapping, so that both forecast and analysis are carried out entirely in latent space; on the reported nonlinear PDE experiment, the relative error was $R$00 for LAE-EnKF versus $R$01 for EnKF, with online computational times $R$02 s for LAE-EnKF and $R$03 s for EnKF (Tong et al., 6 Mar 2026). AD-EnKF makes the ensemble Kalman filter end-to-end differentiable and uses automatic differentiation through the filtering recursion to learn dynamics, outperforming EM- and PF-based alternatives on Lorenz-96 while retaining EnKF scalability in high dimensions (Chen et al., 2021).

Another branch imports adaptation mechanisms from outside filtering. The KFLS/GF-RLS line shows that the Kalman filter is the recursive minimizer of a generalized least-squares cost and uses RLS-style forgetting inside the covariance update to obtain adaptive Kalman filters. In the reported mass-spring-damper example with intermittent, unmodeled collisions, the adaptive filter sharply dropped its forgetting factor after impacts, increased effective covariance, and reduced displacement and velocity estimation error relative to a fixed-covariance KF (Lai et al., 2024). On differentiable manifolds, IKFoM formulates predict and update using $R$04, $R$05, and $R$06, thereby separating manifold constraints from system behavior; the reported lidar–inertial case study showed drifts of $R$07 m over $R$08 m, $R$09 m over $R$10 m, and $R$11 m over $R$12 m, with real-time runtime comparable to hand-engineered counterparts (He et al., 2021).

A further generalization studies randomized sensing rather than adaptive update structure. In continuous-time Kalman-Bucy filtering with random measurement matrices and noise covariances, the expected covariance $R$13 satisfies the closed-form upper bound

$R$14

where $R$15 solves a deterministic Riccati equation driven by the averaged information matrix $R$16. Applied to spatiotemporal field estimation, this leads to a grid-independent lower bound on expected clarity and exposes a composite sensing parameter combining the number of sensors, noise level, and measurement period (Wang et al., 29 Oct 2025). This suggests that the “evolving” aspect of Kalman methodology also includes stochastic sensing geometry and pre-deployment design limits, not only adaptive estimation rules.

Across these strands, one recurring misconception is that evolution in Kalman filtering must imply a black-box departure from interpretability. The literature reviewed here points in the opposite direction. Kalman Evolve in the narrow sense uses compact symbolic programs (Saketos et al., 26 May 2026); adaptive learned filters keep the gain as the main learned object (Ni et al., 2023); manifold and latent-space formulations expose the geometry or representation in which the recursion is valid again (He et al., 2021, Tong et al., 6 Mar 2026). The broader significance is therefore methodological rather than doctrinal: Kalman filtering remains the organizing recursion, but the objects that enter that recursion—gain, covariance, coordinates, sensor model, or update law—are increasingly treated as adaptive, learned, or discoverable.