Spectral Distillation: From Nonlinear Dynamics to Linear State-Space Models

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.

UpSkill: Mutual Information Skill Learning for Structured Response Diversity in LLMs

Reinforcement Learning with Verifiable Rewards (RLVR) has improved the reasoning abilities of large language models (LLMs) on mathematics and programming tasks, but standard approaches that optimize single-attempt accuracy can inadvertently suppress response diversity across repeated attempts, narrowing exploration and overlooking underrepresented strategies. We introduce UpSkill, a training time method that adapts Mutual Information Skill Learning (MISL) to LLMs for optimizing pass@k correctness. We propose a novel reward that we implement within Group Relative Policy Optimization (GRPO): a token-level mutual information (MI) reward that encourages trajectory specificity to z. Experiments on GSM8K with three open-weight models, Llama 3.1-8B, Qwen 2.5-7B, and R1-Distilled-Qwen2.5-Math-1.5B, show that UpSkill improves multi-attempt metrics on the stronger base models, yielding mean gains of ~3% in pass@k for both Qwen and Llama without degrading pass@1. Additionally, we find both empirical and theoretical evidence that improvements in pass@k are closely tied to the mutual information objective.

SpectraLDS: Provable Distillation for Linear Dynamical Systems

We present the first provable method for identifying symmetric linear dynamical systems (LDS) with accuracy guarantees that are independent of the systems' state dimension or effective memory. Our approach builds upon recent work that represents symmetric LDSs as convolutions learnable via fixed spectral transformations. We show how to invert this representation, thereby recovering an LDS model from its spectral transform and yielding an end-to-end convex optimization procedure. This distillation preserves predictive accuracy while enabling constant-time and constant-space inference per token, independent of sequence length. We evaluate our method, SpectraLDS, as a component in sequence prediction architectures and demonstrate that accuracy is preserved while inference efficiency is improved on tasks such as language modeling.