Dynamical Low-Rank Kalman-Bucy Process

• DLR-KBP is a filtering approach that approximates the full covariance matrix using a low-rank factorization, thereby reducing computational complexity in high-dimensional systems.
• It projects the classical Riccati equation onto the tangent space of a low-rank manifold using Oja’s principal component flow, ensuring efficient state estimation and convergence.
• The algorithm is widely applied in large-scale data assimilation and PDE-based uncertainty quantification, offering robust theoretical guarantees and computational savings.

The Dynamical Low-Rank Kalman-Bucy Process (DLR-KBP) is a class of filtering algorithms designed to address the computational challenges of high-dimensional Kalman-Bucy filtering by propagating low-rank approximations of the state covariance matrix. The DLR-KBP leverages the dynamical low-rank (DLR) methodology, projecting the infinite-dimensional Riccati flow for the covariance onto a time-dependent manifold of low-rank symmetric positive semidefinite matrices. Central to the dynamics are the evolution of the principal subspace using Oja’s principal component flow, and a reduced-order Riccati equation for the low-dimensional covariance factor. This makes DLR-KBP amenable to large-scale estimation, data assimilation, and state-space modeling under both continuous and discrete observation schemes (Nobile et al., 14 Sep 2025, Schmidt et al., 2023, Tsuzuki et al., 2024, Bonnabel et al., 2012).

1. Mathematical Structure and Model Setting

DLR-KBP operates within the framework of a linear time-invariant (LTI) stochastic system described by

$$dx_t = A x_t\,dt + B\,dw_t,\qquad dy_t = C x_t\,dt + D\,dv_t,$$

where $w_t, v_t$ are independent standard Wiener processes, with covariance matrices $Q = B B^\top$ and $R = D D^\top$ for process and observation noise, respectively. The classical Kalman-Bucy filter propagates the state estimate $\hat x_t$ and its error covariance $P_t$ by

$$d\hat x_t = A\hat x_t\,dt + P_t C^\top R^{-1} \left( dy_t - C\hat x_t\,dt \right),\qquad \frac{d}{dt}P_t = A P_t + P_t A^\top + Q - P_t C^\top R^{-1} C P_t.$$

For high-dimensional systems, maintaining and evolving the full $n \times n$ covariance matrix is computationally prohibitive. DLR-KBP addresses this by seeking a low-rank factorization,

$P_t \approx U_t S_t U_t^\top,$

with $U_t \in \mathbb{R}^{n \times r}$, $U_t^\top U_t = I_r$, and $S_t$ symmetric positive definite, with $r\ll n$ (Tsuzuki et al., 2024, Nobile et al., 14 Sep 2025, Schmidt et al., 2023).

2. Low-Rank Projection and Evolution Equations

The DLR-KBP applies a geometric projection of the filter Riccati operator onto the tangent space of the low-rank manifold. The projection operator at $P = U S U^\top$ is given by

$$\mathcal{P}_U(M) = U U^\top M + M U U^\top - U U^\top M U U^\top,$$

where $M$ is a symmetric matrix. The projected Riccati equation yields the coupled evolution: $$\begin{aligned} \dot U_t &= (I - U_t U_t^\top) A U_t, \ \dot S_t &= U_t^\top A U_t S_t + S_t U_t^\top A^\top U_t + U_t^\top Q U_t - S_t C_u^\top R^{-1} C_u S_t, \end{aligned}$$ where $C_u = C U_t$, and the process noise $Q$ is typically restricted to the leading subspace (projected or low-rank form). For the filtering mean, the update is

$$d\tilde{x}_t = (A - U_t S_t U_t^\top C^\top R^{-1}C) \tilde{x}_t\,dt + U_t S_t U_t^\top C^\top R^{-1} dy_t.$$

The update structure guarantees that both mean and covariance remain in the reduced-order subspace across time (Tsuzuki et al., 2024, Nobile et al., 14 Sep 2025, Bonnabel et al., 2012).

3. Geometry, Stability, and Error Bounds

The manifold of rank-$r$ positive semidefinite (PSD) matrices,

$$\mathcal{M}_r = \{ Y = U S U^\top : U \in \operatorname{St}(n, r),\, S \in \mathbb{S}_{++}^r \},$$

is equipped with the geometric “sum-of-parts” Riemannian metric, capturing both subspace (Stiefel manifold) and shape (PSD cone) contributions (Bonnabel et al., 2012). The Oja subspace flow,

$\dot U = (I_n - U U^\top)A U,$

preserves the Stiefel constraint and, for generic initializations, converges to the $r$-dimensional invariant subspace associated with the eigenvalues of $A$ with largest real part. Local stability is proved for the equilibrium corresponding to this dominant subspace; domain-of-attraction results provide sufficient initial misalignment bounds for convergence (Tsuzuki et al., 2024). Once the subspace has converged, the reduced Riccati equation yields a unique steady state provided controllability and observability are inherited.

Error analysis in the small-noise regime ($\Sigma \approx 0$) demonstrates that the DLR-KBP estimate remains near-optimal, with Gronwall-type bounds ensuring that the low-rank mean and covariance errors remain controlled for all times: $$\|P_t^{\mathrm{DLR}} - P_t\|_F \leq C(T) ( \|P_0^{\mathrm{DLR}} - P_0\|_F + \varepsilon ),$$ when the neglected noise component is sufficiently small (Nobile et al., 14 Sep 2025).

4. Algorithmic Implementation and Numerical Aspects

DLR-KBP propagates the low-rank factors using explicit time integration (e.g., Euler–Maruyama), with periodic re-orthonormalization of $U_t$. Efficient integrators such as the “Basis-Update–Galerkin” (BUG) scheme provide numerical stability, particularly when singular values of the covariance factor become small (Schmidt et al., 2023). Each iteration involves:

• Advancing $U_n$ via $$\tilde U_{n+1} = U_n + \Delta t (I - U_n U_n^\top) A U_n$$, followed by QR-reorthonormalization.
• Updating $S_{n+1}$ with projected Riccati terms.
• Updating the mean vector $m_t$ accordingly.

Assimilation steps (discrete observation updates) can be realized by low-rank variants of the Kalman update, maintaining the rank structure and achieving $O(n r^2 + m r^2 + r^3)$ complexity per time step under fast operator assumptions (Schmidt et al., 2023).

A summary of per-step computational costs:

Step	Standard KBP	DLR-KBP
Full Riccati update	$O(n^3)$	$O(n r^2 + r^3)$
Measurement update	$O(n^2 m)$	$O(m r^2 + n r^2)$

For $r\ll n$, DLR-KBP provides substantial computational savings while retaining estimation accuracy in low-noise, low-intrinsic-dimensionality regimes (Nobile et al., 14 Sep 2025, Schmidt et al., 2023).

5. Theoretical Guarantees and Rank Selection

DLR-KBP maintains rigorous stability properties under appropriate conditions. If the system is controllable and observable, and the low-rank dimension $r$ matches or exceeds the number of unstable or weakly stable modes (number of eigenvalues of $A$ with real part $\geq 0$), then the closed-loop dynamics are stabilized: $A - P_r C^\top R^{-1} C$ has all its eigenvalues in the left-half plane, ensuring a bounded estimation error covariance. This condition forms the basis for practical rank selection: count the number $r'$ of modes with real part $\geq 0$ and set $r \geq r'$ to guarantee Hurwitz stability and optimality commensurate with the full Kalman-Bucy filter (Tsuzuki et al., 2024).

6. Relation to Ensemble and Hybrid Methods

Extensions of DLR-KBP to ensemble-based filtering, notably the DLR-Ensemble Kalman Filter (DLR-EnKF), enable propagation of large ensembles in the low-dimensional dominant subspace. Propagation-of-chaos results establish convergence of the empirical ensemble to the DLR-KBP mean-field limit at $O(P^{-n/2})$, allowing reduced Monte Carlo error at fixed computational cost compared to standard EnKF (Nobile et al., 14 Sep 2025). DLR-KBP and DLR-EnKF are thus especially effective in regimes where the filter covariance is low-rank and the effective signal dimension is small.

7. Applications, Limitations, and Extensions

DLR-KBP is particularly suited for large-scale data assimilation (e.g., atmospheric or oceanographic state estimation), high-dimensional spatio-temporal regression, and PDE-based uncertainty quantification where the Kalman-Bucy filter is otherwise infeasible. Its main limitation is the assumption that noise and uncertainty are confined predominantly to a low-dimensional subspace; full-rank noise degrades the accuracy of the low-rank approximation. Extensions under investigation include incorporation of hyper-reduction, structure-preserving time integrators, and techniques for localization and inflation in very high-dimensional ensemble filtering settings (Nobile et al., 14 Sep 2025).

8. Summary Table of Key Features

Feature	DLR-KBP	Reference
Covariance factorization	$P_t \approx U_t S_t U_t^\top$	(Tsuzuki et al., 2024)
Subspace evolution	Oja flow, $\dot U = (I-UU^T)AU$	(Tsuzuki et al., 2024, Bonnabel et al., 2012)
Reduced Riccati dynamics	$$\dot S = A_u S + S A_u^\top + Q_u - S C_u^\top R^{-1} C_u S$$	(Tsuzuki et al., 2024)
Computational cost per step	$O(n r^2 + r^3)$, dominant for $r\ll n$	(Nobile et al., 14 Sep 2025, Schmidt et al., 2023)
Stability/Boundedness	Provided $r \geq r'$, controllable/observable	(Tsuzuki et al., 2024)
Ensemble extension	DLR-EnKF, with propagation-of-chaos	(Nobile et al., 14 Sep 2025)

DLR-KBP offers a mathematically grounded and computationally efficient solution to high-dimensional filtering when signal and uncertainty are concentrated in a low-dimensional manifold, with robust theoretical guarantees for mean and covariance fidelity, provided the rank and initialization conditions are met (Tsuzuki et al., 2024, Nobile et al., 14 Sep 2025, Schmidt et al., 2023, Bonnabel et al., 2012).