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.
Many decentralized distillation methods are designed around training-time coordination, yet deploy each node in isolation even when more capable neighbors remain available at inference time. This is an incomplete objective for settings such as IoT, where devices are heterogeneous, data is scarce and skewed, and a node's strongest neighbors may far exceed its own local capacity. We study how nodes should train so that their predictions compose well at deployment, and how each node should learn whom to trust. Under a server-free, model-agnostic protocol where nodes exchange only queries and soft predictions, we propose Learned Neighbor Trust (LNTrust) wherein each node learns a compact trust function over its neighborhood from local validation evidence. This trust function gates auxiliary distillation during training and defines a deployment ensemble at inference, so that collaboration learned during training transfers directly to deployment. Across datasets and topologies, LNTrust improves deployed accuracy over the strongest output-only baseline by large margins while using significantly less communication than previous methods.