State-Conditioned Spectral Propagation for Closed-Form Continuous-Time Liquid Neural Networks

Irregular observation sequences exhibit concurrent short-term fluctuations and long-term trends. A key challenge is to equip hidden states with multiple adaptive decay timescales while preserving closed-form propagation. We propose the State-Conditioned Spectral Closed-Form Continuous-Time Network (SC-SCfC), which models hidden-state evolution toward an interval target between consecutive observations as a mixture of multiple exponential decays. While individual state components can blend fast and slow decay rates, restricting decay to fixed coordinate axes limits the model's expressiveness. To overcome this, we further propose a state-conditioned matrix allocation (SC-MA) mechanism that extends SC-SCfC. By decomposing the residual between the current state and its target into vector components in the latent space using positive semi-definite allocation matrices that sum to the identity, each component can decay at its own rate along data-dependent directions. The parameters remain fixed within each interval, yielding closed-form propagation. Experiments across four benchmark families demonstrate competitive performance in classification and remaining-useful-life regression.