Technical Summary: Spectral Distillation from Nonlinear Dynamics to Linear State-Space Models

Problem Statement

The central challenge addressed by this work is the learning of nonlinear dynamical systems. While Linear Dynamical Systems (LDS) offer compact state-space representations, efficient simulation, and rich spectral theory for analysis, most real-world systems are inherently nonlinear. Learning a nonlinear state-space model, or even finding its best linear approximation, typically requires solving a fundamentally non-convex system identification problem, which is often fragile and computationally difficult.

The paper asks whether it is possible to extract a simple, faithful linear dynamical system representation from observations of an unknown nonlinear system without ever solving a non-convex identification problem.

Methodology

The authors propose a two-stage, end-to-end provable pipeline that separates the learning phase from the representation phase. The process converts observations from a nonlinear system into a compact, explicit recurrent LDS using only convex optimization followed by a structured distillation step.

1. Convex Learning via Observation Spectral Filtering (OSF)

The first stage employs Observation Spectral Filtering (OSF), a convex method that learns an implicit spectral predictor.

• Mechanism: Instead of attempting to recover latent states or fit a nonlinear model directly, OSF operates in a fixed temporal feature space derived from the Hankel matrix. It uses a set of fixed spectral filters (geometric responses) to convolve past inputs and observations.
• Optimization: The predictor parameters (readout matrices) are learned via squared loss, which remains convex because the features are fixed.
• Competitor: OSF competes with the best finite-dimensional linear observer (Luenberger observer) for the system. Its performance guarantee depends on the Luenberger complexity ($Q_\star$) of the best observer—a measure of observability and spectral gap—rather than the dimension of the latent nonlinear state.

2. Spectral-to-LDS Distillation

The second stage converts the implicit, convolutional OSF predictor into an explicit, recurrent state-space model.

• Mechanism: Leveraging recent results on spectral representations (SpectraLDS), the method approximates the learned spectral filters using a finite set of geometric filters (recurrent states with poles $\alpha_i$).
• Realization: The convolutional features of the OSF predictor are replaced by linear combinations of recurrent states. This transforms the predictor into a standard LDS with a transition matrix defined by the chosen poles and a mixing matrix.
• Result: The output is a compact recurrent model where memory and per-step computation costs depend only on the observation and output dimensions, not on the potentially massive dimension of a nonlinear lift.

Key Contributions

1. A Nonlinear-to-LDS Distillation Pipeline

The paper presents the first end-to-end method for extracting a "best-in-hindsight" LDS representation of nonlinear dynamics through convex learning followed by provable distillation. The pipeline consists of:

1. Learning a predictor via OSF in a convex spectral space.
2. Distilling this predictor into an explicit recurrent LDS.

2. Dimension-Free Prediction Guarantee

The main theoretical result (Theorem 3.1) provides a bound on the average prediction error of the distilled LDS. The error decomposes into two terms:
$$\frac{1}{T} \sum_{t=1}^T \|y_t - \hat{y}^{LDS}_t\|^2 \leq \tilde{O}\left( \frac{Q_\star^2 \log(Q_\star) \log^7 T}{\sqrt{T}} + \lambda_{\max}(F)^2 \exp\left(-\frac{2k}{\log T}\right) \right)$$

• OSF Learning Term: Governed by the Luenberger complexity $Q_\star$ of the best observer. This term is independent of the latent dimension of the nonlinear system.
• Distillation Term: An exponentially small error term resulting from approximating the spectral filters with recurrent geometric responses. This term depends on the number of spectral filters $k$ and the number of LDS modes $h$.
• Significance: The guarantee is dimension-free, depending on observer complexity rather than the latent dimension required to represent the nonlinear system.

3. Explicit Compact Representation

Unlike the implicit OSF predictor, the final output is a standard LDS with $O(h \cdot (d_{in} + d_{out}))$ memory and per-step simulation cost. This makes the model suitable for long-horizon simulation and control.

4. Empirical Validation

The authors validate the "train-then-distill" approach on two fronts:

• Linear LDS Benchmarks: On synthetic data with long memory (slowly decaying impulse responses), the distilled LDS outperforms directly trained Diagonal and General LDS baselines, often by orders of magnitude, and matches or exceeds Linear FIR baselines with significantly fewer parameters.
• MuJoCo Behavior Cloning: On nonlinear control tasks (Walker2d, Hopper, HalfCheetah, Humanoid, Ant, Pusher), the distilled LDS achieves higher or comparable episode rewards compared to directly trained Diagonal LDS baselines, despite having fewer parameters than the teacher policy.

Significance and Claims

The paper claims to provide a principled route from nonlinear observations to compact linear state-space models. Its primary significance lies in demonstrating that:

1. Convexity is sufficient: One can learn complex nonlinear dynamics effectively by first learning a convex spectral predictor and then distilling it, avoiding the pitfalls of non-convex system identification.
2. Optimization vs. Expressivity: The performance gains observed in experiments appear to stem from the convexity of the spectral parameterization (which facilitates better optimization) rather than from a richer model class, as the final deployed models are standard LDSs.
3. Theoretical Rigor: It offers the first non-asymptotic, dimension-free guarantees for extracting linear representations from nonlinear data, linking the error directly to the observability properties of the system.

The authors note that while the theoretical bound controls average one-step prediction error under teacher forcing, extending these guarantees to closed-loop rollout error (where errors compound) remains an open direction for future work.