The Hilbert–Schmidt Koopman Operator: At the Heart of Spectral LLMs

This paper introduces a proposed architecture for language models in which the internal state is represented as a structured operator rather than a conventional hidden-state vector. Instead of treating each inference step as a point in a high-dimensional embedding space, the model evolves a constrained operator that preserves interpretable spectral properties throughout the computation.

The architecture combines concepts from operator theory, Hilbert–Schmidt geometry, Koopman-style dynamical systems, and spectral analysis to create a latent representation whose evolution can be monitored using intrinsic quantities such as spectral energy, entropy, effective rank, and similarity. By maintaining these operator-level invariants, the framework aims to provide a mathematically structured view of how information evolves during inference.

A central contribution is the introduction of a bounded operator manifold that constrains every state update to remain within a well-defined geometric space. This enables the latent state to be compared using Hilbert–Schmidt similarity, allowing previous operator states to be organised into a non-parametric memory repository. Rather than relying solely on learned parameters, this memory mechanism supports the identification, merging, and retrieval of related concepts through operator similarity.

The paper also proposes a spectral decoding strategy, in which predictions are generated from invariant spectral characteristics of the operator rather than directly from raw hidden-state coordinates. This provides an alternative interpretation of internal representations while preserving mathematical consistency across successive updates.