Extraction of linearized models from pre-trained networks via knowledge distillation

Can nonlinear dynamical systems be learned through a compact linear state-space representation, without directly solving a non-convex system-identification problem? We give a provable pipeline for doing so. Starting from observations of an unknown nonlinear dynamical system, we first learn an implicit spectral predictor using Observation Spectral Filtering (OSF), a convex method that competes with the best linear observer for the system. We then apply spectral-to-LDS distillation to convert this predictor into an explicit recurrent linear dynamical system. Our main theorem shows that the average prediction error of the distilled LDS decomposes into an exponentially-small distillation term and the OSF learning term governed by the Luenberger complexity of the best observer. The guarantee is dimension-free: it depends on observer complexity rather than on the latent dimension needed to represent the nonlinear system. To our knowledge, this yields the first end-to-end provable method for extracting a best-in-hindsight LDS representation of nonlinear dynamics through convex learning followed by provable distillation. Experiments on linear LDS benchmarks and MuJoCo behavior cloning show that the train-then-distill pipeline produces compact LDS predictors that match or outperform directly trained baselines.

Dataset distillation compresses a large training set into a small synthetic set that preserves downstream training utility. While most existing methods target training networks from scratch, modern visual transfer learning often uses frozen pre-trained encoders followed by lightweight linear probing. Existing distillation methods for this setting either unroll iterative linear-probe updates with trajectory-based gradient matching, or rely on closed-form formulations originally designed for from-scratch training with neural-tangent-kernel (NTK) approximations. Neither route exploits the fact that frozen-feature linear probing admits a closed-form solution determined directly by the pre-trained features themselves, with no infinite-width approximation and no inner-loop trajectory. We propose Closed-Form Linear-Probe Dataset Distillation (CLP-DD), a bilevel formulation that computes the linear probe induced by the synthetic set with a sample-space kernel ridge solver. The synthetic images are then updated by evaluating this induced classifier on real features through a temperature-scaled softmax cross-entropy, where the classifier columns act as learned class anchors in feature space. We further show that the choice of outer objective is decisive: pairing the closed-form inner solver with a standard MSE outer loss substantially underperforms trajectory-based methods, while the discriminative outer loss closes most of the gap. On ImageNet-100 with four pre-trained backbones, CLP-DD substantially improves over LGM without DSA and approaches LGM with DSA at a fraction of the computational cost. On ImageNet-1K, CLP-DD matches or surpasses LGM with DSA on three of four backbones while running roughly $14\times$ faster and using less than one-eighth of the GPU memory.

This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require ``anti-collapse mechanisms'' to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.