Unobservable Subspace Evolution (USE)

• Unobservable Subspace Evolution (USE) is a framework that formalizes the tracking of latent, unobservable subspaces in dynamic systems affected by symmetry and limited observations.
• USE integrates symmetry reduction and alignment techniques in filtering, visual-inertial navigation, subspace clustering, and streaming estimation to improve estimation consistency.
• Practical implementations employ transformation-based and re-evaluation-based alignment methods to restore estimator performance and mitigate misalignment issues.

Unobservable Subspace Evolution (USE) refers to the dynamical characterization, tracking, and analysis of system components or latent variable structures that, due to intrinsic symmetries, observational limitations, or incomplete data, remain unobservable to direct measurement. USE formalizes the evolution of such subspaces within filtering, estimation, and clustering contexts, detailing how their structure interacts with inference algorithms, affects estimator consistency, and drives the need for principled alignment or reduction techniques. This framework finds relevance in geometric symmetry reduction for Kalman filtering, visual-inertial navigation, evolutionary subspace clustering, and high-dimensional streaming subspace estimation.

1. Formal Definition and Contexts

USE denotes the explicit tracking of unobservable subspaces—collections of system states or data substructures invariant to the observation process or induced by group symmetries. Let $x$ be the system state evolving according to $x_{k+1}=F(x_k,u_k)$, observed as $z_k=H(x_k)$. If $x$ admits a group action $\varphi_g$ with invariance $H(\varphi_g x)=H(x)$, directions associated with the group $G$ are strictly unobservable.

In visual-inertial navigation, the physical unobservable subspace is spanned by the global position and yaw directions; for $N$ visual features, the unobservable subspace is $\mathbb N(x)=[N\, N(x)]$ ($N\in\mathbb R^{N\times3}$ encodes global position, $N(x)\in\mathbb R^{N\times1}$ global yaw) (Tian et al., 22 Nov 2025). In clustering time-series data, unobservable subspaces may correspond to latent bases $U_k(t)$ of subspaces $S_k(t)$, which evolve but are never seen directly (Bai et al., 2021). Streaming matrix completion under missing data similarly features evolving subspace estimates governed by principal angles implicit in incomplete observations (Wang et al., 2018).

2. Symmetry-Induced USE in Filtering

Systems invariant under group actions (SE(2) for planar robots, SE(3) for general pose) yield natural splits between unobservable and observable components. In the Shen & Leok reduction (Shen et al., 2019), the state $$x=(p,\theta,p^f_1,\ldots,p^f_N) \in SE(2)\times\mathbb R^{2N}$$ decomposes globally as $x=(x_N,x_O)$, $x_N=(p,\theta)$ (unobservable, robot pose) and $x_O=(z_1,\ldots,z_N)$ (observable, relative landmark positions), where $z_i=R(\theta)^T(p^f_i-p)$. Infinitesimal generators $\xi\in\mathfrak{se}(2)$ specify the unobservable direction, forming $U_x=\{\xi_X(x):\xi\}$.

Kalman and Bayesian updates, when symmetry-reduced to the observable subspace, proceed as:

• Propagation: block-diagonal on $(x_N,x_O)$.
• Measurement update: acts only on $x_O$; derivative $dh=[0, I]$ ensures $dh\cdot U_x=0$ and no Fisher information leaks into unobservable directions. This guarantees filter consistency and the absence of spurious coupling, even under large measurement noise (Shen et al., 2019).

3. Stepwise Evolution, Misalignment, and USA in VINS

USE in estimation pipelines tracks not only dimension but "evaluation-point alignment" of the unobservable subspace. In modern visual-inertial navigation filters:

• Propagation preserves subspace alignment ($\mathbb N(\hat x_k)$).
• Correction steps, including SLAM and MSCKF updates, can leave the estimator's nullspace misaligned: $\mathbf N^+=\mathbb N(\hat x_k)$ after update, instead of $\mathbb N(\hat x_k^+)$.
• A subsequent update almost surely collapses dimension from 4 (physical) to 3 (mismatch), triggering estimator inconsistency (Tian et al., 22 Nov 2025).

USE framework distinguishes three statuses:

1. Mismatched: $\dim(\mathbf N)\ne 4$.
2. Aligned: $\mathbf N=\mathbb N(\hat x^*)$ (current estimate).
3. Misaligned: $\dim(\mathbf N)=4$, $\mathbf N\ne\mathbb N(\hat x^*)$.

Misalignment is the necessary precursor to mismatch and inconsistency.

To remedy, Unobservable Subspace Alignment (USA) is introduced: Transformation-Based USA re-aligns information matrix $\Lambda$ via a rank-one or block transformation $T$ so $T^{-1}\mathbb N(\hat x)=\mathbb N(\hat x^+)$, with minimal Frobenius-norm distance to identity (Tian et al., 22 Nov 2025). Re-evaluation-Based USA updates all Jacobians at the new evaluation point $\hat x^+$. Both restore alignment and prevent collapse, yielding empirical consistency (NEES$\approx$1) across Monte Carlo and real environments (Tian et al., 22 Nov 2025).

4. USE in Evolutionary Subspace Clustering and Latent Time-Series Models

In multi-dimensional time series, the USE problem is formulated as tracking continuously evolving latent subspaces $S_k(t)$ that are unobservable; only the sampled data $X(t_j)$ are observed. The neural ODE model for evolutionary subspace clustering (NODE-ESCM) models the evolution of the subspace self-expressiveness matrix $C(t)$ via $dh/dt=g_\theta(h(t),X(t),t)$, where $h(t)$ encodes the independent entries of $C(t)$ (Bai et al., 2021). Self-expressiveness, $X(t_j)\approx X(t_j)C(t_j)$, is enforced via a sparsity-regularized objective.

ODE solvers interpolate $h(t)$ at arbitrary times, enabling reconstruction, clustering, and affinity extraction even with missing—hence unobservable—data. Identifiability is inherited from sparse subspace clustering guarantees, requiring sufficient separation and incoherence between subspaces, and manageable subspace drift speeds relative to sampling intervals. Empirical tests reveal superior accuracy, clustering performance, and robustness to missing data compared with discrete-time LSTM-ESCM or SSC methods (Bai et al., 2021).

5. Streaming Subspace Estimation and ODE Limits

High-dimensional analysis of streaming algorithms for incomplete data treats the principal angles between true and estimated subspaces as a dynamic process. Oja’s method, GROUSE, and PETRELS evolve subspace estimates $X_k$ via stochastic, possibly partial observations. Acceleration of time ($k=\lfloor nt \rfloor$) reveals that the cosine similarity between true and estimated subspaces, $Q^{(n)}(t)$, converges (as $n\to\infty$) to the solution $Q(t)$ of a deterministic matrix ODE, e.g., $dQ/dt=F(Q,\tau I)$ for Oja/GROUSE, with $F$ capturing intrinsic signal-to-noise and sampling ratio (Wang et al., 2018).

This scaling limit predicts both transient and steady-state evolution of subspace estimates—including phase transitions, finite-sample bounds ($\mathcal O(1/\sqrt n)$ deviation), and trade-offs between missing-data fraction $\alpha$, SNR, and algorithmic step-size. Asymptotic equivalence between distinct update schemes (Euclidean vs. Grassmannian) is demonstrated, as are interpretable thresholds for informative recovery and algorithmic breakdown (Wang et al., 2018). USE here manifests via deterministic flow of latent alignment metrics under high-dimensional random environments.

6. Theoretical Guarantees and Limitations

Existence and uniqueness of the USE trajectory stem from Lipschitz and continuity conditions on the governing ODEs, e.g., Picard–Lindelöf theory for neural ODE parameterizations (Bai et al., 2021). Consistency of numerical solvers, alignment-preserving update transformations, and robustness to missing or irregular data are formalized and empirically validated.

Identifiability, however, can be compromised by fast-evolving subspaces (relative to sampling rate), insufficient inter-subspace separation, or excessive model complexity—especially in neural-based formulations. In filtering applications, failure to track and align the evaluation-point of the unobservable subspace leads directly to misalignment and consequential estimator inconsistency (Tian et al., 22 Nov 2025).

7. Applications and Empirical Validation

USE and its associated reduction/alignment methodologies have demonstrated impact across diverse domains:

• Navigation and SLAM: symmetry-reduced Kalman filters outperform standard approaches under high measurement noise, with position/orientation error reductions of 30–50% and improved stability in unobservable directions (Shen et al., 2019).
• Visual-inertial estimation: USA achieves estimator consistency and accuracy matching or exceeding FEJ and RI-EKF at lower complexity (Tian et al., 22 Nov 2025).
• Time-series clustering: NODE-ESCM attains best-in-class accuracy on motion segmentation, ocean water-mass detection, and socio-economic clustering even with missing instants (Bai et al., 2021).
• High-dimensional streaming inference: ODE analysis predicts and bounds algorithmic performance, revealing phase transitions and scaling laws for informative recovery as missing data/step-size and SNR parameters vary (Wang et al., 2018).

Concluding Synthesis

Unobservable Subspace Evolution provides a rigorous framework for understanding, modeling, and mitigating the effects of latent invariant structures in dynamic systems and data analysis. By articulating both the geometric and algorithmic dynamics of these subspaces, USE enables design of robust filtering, clustering, and streaming inference algorithms that remain consistent, interpretable, and resilient under symmetry, incomplete observation, and subspace drift.