Optimized Kalman Smoother (OKS)

• Optimized Kalman Smoother (OKS) is an optimization framework that reformulates traditional Kalman smoothing as a parameter-tuning problem using data-driven and batch MAP approaches.
• It incorporates methods such as Gauss–Newton, full Newton recursions, robust penalties, and orthogonal transformations to enhance performance, sparsity, and numerical efficiency.
• Applications of OKS span from vehicle tracking and digital twins to gravitational-wave analysis, demonstrating its adaptability and scalability in complex state-space models.

Optimized Kalman Smoother (OKS) denotes a class of Kalman-smoothing formulations in which smoothing is treated as an explicit optimization, tuning, or implementation-design problem rather than as a fixed recursion with hand-set parameters. In a narrow usage, OKS refers to learning or tuning the parameters of a linear-Gaussian smoother directly from data by minimizing held-out prediction error (Barratt et al., 2019). In a broader usage, the term also covers batch MAP formulations for nonlinear smoothing, robust and sparse generalized smoothers, and numerically optimized square-root or parallel implementations (Howell et al., 4 Oct 2025, Aravkin et al., 2013, 2207.13526, Gargir et al., 17 Feb 2025).

1. Nomenclature and scope

The literature does not present a single universally fixed definition of OKS. One explicit use of the name appears in work comparing the “Optimized Kalman Smoother (OKS)” against unscented smoothing for nonlinear systems (Howell et al., 4 Oct 2025). A second, closely related usage appears in the data-driven tuning framework of “Fitting a Kalman Smoother to Data,” which develops what can reasonably be called an optimized Kalman smoother because it chooses smoother parameters from held-out predictive performance rather than manual tuning (Barratt et al., 2019). Other papers do not use the exact label, but adopt the same optimization viewpoint for smoothing (Aravkin et al., 2013, Aravkin et al., 2011).

Usage of OKS	Representative papers	Characteristic mechanism
Data-driven tuned smoother	(Barratt et al., 2019)	minimize held-out prediction loss over smoother parameters
Nonlinear batch MAP smoother	(Howell et al., 4 Oct 2025, Yaghoobi et al., 2023)	Gauss–Newton or Newton iterations over the full trajectory
Robust or sparse smoother	(Aravkin et al., 2010, Aravkin et al., 2011, Gao et al., 2019)	replace quadratic penalties by Student’s $t$, PLQ, or $L_1$ terms
Numerically optimized smoother	(2207.13526, Gargir et al., 17 Feb 2025)	orthogonal transformations, sparse QR, or parallel-in-time execution

Taken together, these papers suggest that OKS is best understood as an umbrella concept for optimization-oriented Kalman smoothing. This also clarifies what it is not. A paper may use a standard Kalman filter or smoother in an application without proposing a new OKS method; for example, the causal digital twin bearing study is explicitly useful as a modeling and application reference, but not as a source of a new smoother derivation or optimization method (Madhavan, 2021).

2. Linear-Gaussian foundations and the optimization viewpoint

A core OKS formulation begins from the standard finite-horizon linear state-space model

$x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$

with $w_t\sim \mathcal N(0,W)$ and $v_t\sim \mathcal N(0,V)$, and parameter vector

$\theta=(A,W^{-1/2},C,V^{-1/2})$

as used in the data-fitting framework (Barratt et al., 2019). In that paper, smoothing with missing measurements is written as a constrained least-squares problem:

$$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{t=1}^{T-1} \left\|W^{-1/2}(\hat x_{t+1}-A\hat x_t)\right\|_2^2 + \sum_{t=1}^T \left\|V^{-1/2}(\hat y_t-C\hat x_t)\right\|_2^2 \ \text{subject to} & (\hat y_t)_i = (y_t)_i,\qquad (i,t)\in\mathcal K. \end{array}$$

This is exactly the MAP or maximum-likelihood smoothing problem for the linear-Gaussian model (Barratt et al., 2019).

The broader optimization viewpoint goes back at least to the batch least-squares formulation

$$\min_x \; \frac12 \|Hx-z\|_{R^{-1}}^2 + \frac12 \|Gx-w\|_{Q^{-1}}^2,$$

where the full trajectory is stacked into a single vector and the normal equations are symmetric positive definite and block tridiagonal (Aravkin et al., 2013). A central result of that work is that the classical Rauch–Tung–Striebel smoother is equivalent to applying a block tridiagonal solver to this batch system (Aravkin et al., 2013). This viewpoint is foundational for later OKS variants because it makes the smoother a structured optimizer over an entire trajectory rather than only a recursive estimator.

3. Data-driven parameter tuning

The clearest “tuned from data” OKS is the auto-tuning framework of (Barratt et al., 2019). Its central idea is to hide a subset of observed entries, smooth using the remaining data, and tune smoother parameters to reconstruct the hidden entries accurately. If $\mathcal M\subseteq \mathcal K$ is the held-out mask, the prediction loss is

$$L(\theta)=\sum_{(i,t)\in\mathcal M}\left((\hat y_t)_i-(y_t)_i\right)^2.$$

The full tuning problem is

$L_1$0

with $L_1$1, where $L_1$2 can impose structure such as diagonal $L_1$3 and $L_1$4, nonnegative $L_1$5, or fixed $L_1$6 and $L_1$7, and $L_1$8 can encode regularization or hard constraints (Barratt et al., 2019).

A technical contribution of that paper is differentiating the held-out loss through the smoother solve by implicit differentiation of the KKT system. After the forward smoothing solve, an adjoint linear system with the same sparse KKT matrix is solved, so the gradient is obtained at “little to no additional cost” (Barratt et al., 2019). Parameters are then updated by proximal gradient:

$L_1$9

Because the KKT matrix has chain and banded sparsity, the per-iteration cost scales essentially linearly in the horizon $x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$0 (Barratt et al., 2019).

The two reported demonstrations are population migration in the United States and vehicle smoothing from IMU+GPS data. In the population experiment, over 50 iterations with $x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$1, prediction error dropped from $x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$2 to $x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$3, and test error from $x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$4 to $x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$5, in 31 seconds (Barratt et al., 2019). In the vehicle experiment, running 25 iterations with $x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$6 reduced prediction error from $x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$7 to $x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$8 and test error from $x_{t+1}=Ax_t+w_t,\qquad y_t=Cx_t+v_t,$9 to $w_t\sim \mathcal N(0,W)$0, in 135 seconds (Barratt et al., 2019). The associated implementation is the open-source Python package auto_ks (Barratt et al., 2019).

4. Nonlinear OKS: batch MAP, Gauss–Newton, and Newton recursions

In nonlinear settings, OKS is formulated as a batch MAP problem over the full trajectory. The explicit OKS formulation in (Howell et al., 4 Oct 2025) starts from

$w_t\sim \mathcal N(0,W)$1

and solves

$w_t\sim \mathcal N(0,W)$2

A central claim of that paper is that extended Kalman smoothing can be viewed as one Gauss–Newton step on this objective, whereas OKS continues the Gauss–Newton iterations with Armijo line search (Howell et al., 4 Oct 2025). The paper reports that in linear systems OKS and EKS are identical up to machine precision, while in a highly nonlinear non-harmonic oscillator OKS substantially improves over both EKS and UKS. For example, at parameter set $w_t\sim \mathcal N(0,W)$3, position error is $w_t\sim \mathcal N(0,W)$4 for EKS, $w_t\sim \mathcal N(0,W)$5 for UKS, and $w_t\sim \mathcal N(0,W)$6 for OKS (Howell et al., 4 Oct 2025).

A second line of work replaces Gauss–Newton by full Newton while preserving recursive structure. “A Recursive Newton Method for Smoothing in Nonlinear State Space Models” shows that the full Newton step can be implemented as smoothing on a modified affine model augmented by a pseudo-measurement (Yaghoobi et al., 2023). The curvature correction appears through

$w_t\sim \mathcal N(0,W)$7

where $w_t\sim \mathcal N(0,W)$8 and $w_t\sim \mathcal N(0,W)$9 collect second-derivative terms from the measurement and dynamics models (Yaghoobi et al., 2023). The paper emphasizes automatic differentiation and uses JAX for first and second derivatives (Yaghoobi et al., 2023).

These nonlinear OKS formulations retain the temporal sparsity of classical smoothing. In (Howell et al., 4 Oct 2025), each Gauss–Newton step leads to a block-tridiagonal linear system. In (Yaghoobi et al., 2023), the full Newton Hessian is also block tridiagonal, and the recursive implementation avoids forming a dense batch Hessian. This suggests that the main distinction between nonlinear OKS and classical nonlinear smoothers is not the abandonment of Kalman structure, but the explicit exploitation of that structure inside iterative nonlinear optimization.

5. Robust, sparse, and constrained extensions

A major extension of OKS is the replacement of quadratic penalties by robust or sparsity-promoting penalties. In (Aravkin et al., 2010), the T-Robust smoother models measurement noise with Student’s $v_t\sim \mathcal N(0,V)$0 and Gaussian process noise, while the T-Trend smoother uses Student’s $v_t\sim \mathcal N(0,V)$1 process noise and Gaussian measurement noise. Both are solved by structured iterations that preserve a block-tridiagonal positive definite subproblem. The reported complexity remains $v_t\sim \mathcal N(0,V)$2 (Aravkin et al., 2010). In the linear outlier experiment with $v_t\sim \mathcal N(0,V)$3 and contaminating $v_t\sim \mathcal N(0,V)$4, the MSEs are $v_t\sim \mathcal N(0,V)$5 for KS, $v_t\sim \mathcal N(0,V)$6 for the $v_t\sim \mathcal N(0,V)$7-Laplace smoother, and $v_t\sim \mathcal N(0,V)$8 for the T-Robust smoother (Aravkin et al., 2010).

A broader convex framework is built from piecewise linear quadratic penalties. “A statistical and computational theory for robust and sparse Kalman smoothing” develops a MAP smoother with PLQ losses such as $v_t\sim \mathcal N(0,V)$9, $\theta=(A,W^{-1/2},C,V^{-1/2})$0, Huber, and Vapnik, and proves that the resulting interior-point method maintains complexity

$\theta=(A,W^{-1/2},C,V^{-1/2})$1

under the stated finite-valuedness and nondegeneracy condition $\theta=(A,W^{-1/2},C,V^{-1/2})$2 (Aravkin et al., 2011). The related optimization viewpoint of (Aravkin et al., 2013) extends smoothing to nonlinear models, linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for, while preserving the computational efficiency associated with block-tridiagonal structure.

Sparsity can also be imposed by variable splitting. In (Gao et al., 2019), the regularized objective

$\theta=(A,W^{-1/2},C,V^{-1/2})$3

is solved by KS-ADMM or IEKS-ADMM, where the smoother handles the quadratic subproblem and soft-thresholding handles the $\theta=(A,W^{-1/2},C,V^{-1/2})$4 term (Gao et al., 2019). In the linear case, the paper gives global convergence under detectability; in the nonlinear case, convergence is local (Gao et al., 2019).

A different robust interpretation appears in (Ruckdeschel et al., 2012), where robustness is defined distributionally for additive outliers and innovation outliers. The AO-robust smoother is recommended empirically, while the IO-robust smoother is reported as less convincing and in need of further research (Ruckdeschel et al., 2012).

6. Numerical linear algebra and high-performance implementations

Another branch of OKS research optimizes the smoother numerically rather than statistically. “UltimateKalman” reformulates linear Kalman filtering and smoothing as a weighted least-squares problem solved incrementally by orthogonal transformations (2207.13526). Its general model is

$\theta=(A,W^{-1/2},C,V^{-1/2})$5

which allows varying state dimensions, varying numbers of observations, missing observations, and unknown initial-state expectation (2207.13526). The implementation supports filtering, prediction, fixed-lag smoothing, and whole-track smoothing. The paper reports a C implementation runtime of about $\theta=(A,W^{-1/2},C,V^{-1/2})$6/step for filtering at $\theta=(A,W^{-1/2},C,V^{-1/2})$7, and smoothing examples of 5 million steps with $\theta=(A,W^{-1/2},C,V^{-1/2})$8 in about 96 s and 100,000 steps with $\theta=(A,W^{-1/2},C,V^{-1/2})$9 in 84 s, both under 16 GB memory (2207.13526).

Parallel runtime optimization is developed in “Parallel-in-Time Kalman Smoothing Using Orthogonal Transformations” (Gargir et al., 17 Feb 2025). That paper solves the standard linear smoothing problem

$$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{t=1}^{T-1} \left\|W^{-1/2}(\hat x_{t+1}-A\hat x_t)\right\|_2^2 + \sum_{t=1}^T \left\|V^{-1/2}(\hat y_t-C\hat x_t)\right\|_2^2 \ \text{subject to} & (\hat y_t)_i = (y_t)_i,\qquad (i,t)\in\mathcal K. \end{array}$$0

by a recursive odd-even sparse QR factorization plus selective inversion for covariance recovery (Gargir et al., 17 Feb 2025). The reported implementation, using TBB, achieves speedups of up to 47x on 64 cores, but is 1.8x to 2.5x slower on a single core because it performs more arithmetic than sequential smoothers (Gargir et al., 17 Feb 2025). The paper also states that the new algorithm is faster and scales better than the parallel Kalman smoother proposed by Särkkä and García-Fernández in 2021 (Gargir et al., 17 Feb 2025).

A lighter-weight real-time example appears in lidar-inertial odometry. FR-LIO uses a tightly-coupled iterated error-state Kalman smoother over lidar sub-frames spanning at least three full scans, with RTS-style backward smoothing and reported “Backward smooth + constraint integrity analysis” times of 0.53 ms, 0.65 ms, and 0.52 ms on three sequences (Liu et al., 2023). This suggests that OKS ideas can also be implemented as local fixed-lag refinement rather than as large batch solves.

7. Applications, limitations, and common misconceptions

Application papers show that OKS-style methods are not confined to generic benchmarks. In causal digital twin estimation for connected bearings, an SVAR causal model is recast as the state-space system

$$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{t=1}^{T-1} \left\|W^{-1/2}(\hat x_{t+1}-A\hat x_t)\right\|_2^2 + \sum_{t=1}^T \left\|V^{-1/2}(\hat y_t-C\hat x_t)\right\|_2^2 \ \text{subject to} & (\hat y_t)_i = (y_t)_i,\qquad (i,t)\in\mathcal K. \end{array}$$1

with a 24-dimensional latent state of time-varying causal factors estimated by a Kalman filter and smoother (Madhavan, 2021). That paper reports that the previous neural network algorithm and Kalman smoother produced very similar results, and that Kalman may perform better for noisy industrial IoT data, but it does not propose a modified or optimized smoother (Madhavan, 2021). In spaceborne gravitational-wave data analysis, a three-state local Taylor model is combined with Kalman filtering, RTS smoothing, and EM-based tuning of $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{t=1}^{T-1} \left\|W^{-1/2}(\hat x_{t+1}-A\hat x_t)\right\|_2^2 + \sum_{t=1}^T \left\|V^{-1/2}(\hat y_t-C\hat x_t)\right\|_2^2 \ \text{subject to} & (\hat y_t)_i = (y_t)_i,\qquad (i,t)\in\mathcal K. \end{array}$$2 and $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{t=1}^{T-1} \left\|W^{-1/2}(\hat x_{t+1}-A\hat x_t)\right\|_2^2 + \sum_{t=1}^T \left\|V^{-1/2}(\hat y_t-C\hat x_t)\right\|_2^2 \ \text{subject to} & (\hat y_t)_i = (y_t)_i,\qquad (i,t)\in\mathcal K. \end{array}$$3, with the EM iteration count set empirically to 5; the method mitigates parameter-estimation biases from data gaps, but degrades for long gaps near coalescence time $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{t=1}^{T-1} \left\|W^{-1/2}(\hat x_{t+1}-A\hat x_t)\right\|_2^2 + \sum_{t=1}^T \left\|V^{-1/2}(\hat y_t-C\hat x_t)\right\|_2^2 \ \text{subject to} & (\hat y_t)_i = (y_t)_i,\qquad (i,t)\in\mathcal K. \end{array}$$4 (Shen et al., 3 Jul 2025). In evolutionary inverse problems, an augmented iterated Kalman smoother includes time-dependent coefficients in the state vector and updates slowly varying mean temperature and conductivity by averaging the smoother estimates (Riedel, 2018).

A common misconception is to treat OKS as a single named algorithm. The papers summarized here suggest a more heterogeneous picture. In some works, “optimized” means direct hyperparameter tuning from held-out prediction error (Barratt et al., 2019). In others, it means batch MAP refinement for nonlinear smoothing (Howell et al., 4 Oct 2025), full Newton recursion (Yaghoobi et al., 2023), robust or sparse reformulation (Aravkin et al., 2010, Aravkin et al., 2011), or numerical optimization of the linear algebra through orthogonal transformations and parallel sparse QR (2207.13526, Gargir et al., 17 Feb 2025). Another misconception is that any application of a Kalman smoother is automatically an OKS; the causal digital twin paper is explicitly valuable as a state-space template and industrial use case, but not as a source of a new smoother derivation or optimization methodology (Madhavan, 2021).

The main limitations recur across these formulations. The outer optimization is often nonconvex in nonlinear settings, so initialization matters and global optimality is not claimed (Barratt et al., 2019, Howell et al., 4 Oct 2025). Robust and sparse models improve fit under outliers or abrupt changes, but add tuning and can require specialized solvers (Aravkin et al., 2010, Aravkin et al., 2011). Numerically optimized implementations improve conditioning, flexibility, or parallel depth, but may incur higher single-core cost or require nonsingular covariance structures (2207.13526, Gargir et al., 17 Feb 2025). The resulting picture is not of a monolithic smoother, but of a research program: Kalman smoothing treated as an optimization problem whose statistical objective, parameterization, regularization, and numerical realization can all be redesigned.