Dynamics-Aware Sampling and Recovery for Tensor-Valued Dynamical Systems

Abstract

We study sampling and recovery for tensor-valued dynamical systems evolving under the t-product, using partial spatial observations over time. Although the t-product Fourier transform yields slicewise matrix dynamics, recovery does not generally decouple across slices. Eigenvalues shared across Fourier-domain frontal slices couple the corresponding eigenspaces into aggregated spectral blocks, which determine the tensor-specific recovery geometry and space--time tradeoff. We establish deterministic identifiability conditions for the initial tensor state. In the diagonalizable case, these are frame conditions on the aggregated blocks. We extend them to non-diagonalizable systems through generalized eigenspaces and Jordan structure. A temporal--spectral factorization of the finite-horizon stacked recovery system underlying Tensor Dynamical Sampling (DS) yields a decoupled two-stage formulation once the temporal threshold is reached. Under independent spatial subsampling, we prove high-probability recovery guarantees using empirical Gram matrices on the aggregated blocks. The bounds depend on aggregated multiplicities, incoherence, and population conditioning. Stability estimates separate the spatial frame factor from the temporal Vandermonde factor and show that increasing temporal depth may worsen numerical conditioning even after identifiability is ensured. We implement Tensor DS and evaluate it on synthetic and real data. Phase transitions track the predicted blockwise sampling thresholds, and stability experiments illustrate Vandermonde conditioning. Experiments on sea surface temperature and turbulence velocity data show that a learned t-product system operator generally yields more accurate and structurally faithful reconstructions than vectorized low-rank, Tucker-type, graph-regularized, graph-spectral, and vectorized dynamical sampling baselines.