Study Shows Two-Layer Linear Auto-Regressive Models Learn to Approximate Kalman Filtering
Researchers demonstrated that two-layer linear auto-regressive models trained on partially observed linear dynamical systems naturally learn representations equivalent to optimal Kalman filtering, without explicit knowledge of the underlying dynamics. This finding addresses a theoretical gap in understanding how auto-regressive models learn latent representations for sequential data. The result provides finite-sample guarantees and suggests auto-regressive models may implicitly discover classical optimal filtering techniques.
A new theoretical analysis shows that two-layer linear auto-regressive models, when trained via empirical risk minimization on data from partially observed linear dynamical systems, converge to approximating Kalman filtering—the classical optimal state estimation technique. The researchers established three key insights: the Kalman filter can be well-approximated by an auto-regressive model with bounded error, the optimization landscape for two-layer models is benign (containing only strict saddles or global minima despite non-convexity), and finite-sample guarantees exist for prediction error, parameter estimation, and latent state recovery. The learned hidden representations coincide with Kalman filter state estimates up to similarity transformations, even though the model receives no explicit information about system dynamics. Numerical simulations corroborate the theoretical predictions, suggesting that auto-regressive models may implicitly discover classical optimal filtering principles during training.
What's missing
The study's own limitations and open questions include: whether these results extend to nonlinear dynamical systems or higher-layer architectures; the practical implications for real-world sequential modeling tasks where data may violate linear assumptions; and how the theoretical guarantees scale with system dimensionality and observation noise levels.
What different sources said
- arXiv cs.AICenter
Two-Layer Linear Auto-Regressive Models Estimate Latent States
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