Finite-sample bounds for non-uniform embedding quality

Derive explicit upper and lower finite-sample bounds on the embedding quality of stable linear Takens delay-coordinate maps constructed from unevenly spaced delays, including an explicit conditioning term analogous to the lag-dependent term for uniformly sampled delays.

Background

The paper extends stable linear Takens embedding results from uniformly spaced delays to non-uniformly spaced delays. It proves an asymptotic stability result as the number of delays tends to infinity, but the corresponding finite-delay estimates are not obtained.

For uniformly sampled delays, geometric sums allow the authors to derive an explicit lag-dependent conditioning term. With non-uniform delays, that geometric structure is lost, preventing the derivation of comparable enveloping bounds and, consequently, a rigorous finite-sample rate of convergence for the embedding quality.

References

Due to the unevenly-sampled delay structure of generalized Vandermonde matrices $T$, it is unknown whether explicit enveloping upper and lower bounds on the embedding quality can be derived. What is possible is to capture the asymptotic bounds for stability as $\ell \to \infty$ to show that $$ can approach isometry.

— Stable Takens' Embedding Theorem for Non-Uniformly-Sampled Linear Systems  (2608.14001 - Ng et al., 14 Aug 2026) in Section IV, “Non-Uniformly-Sampled Takens’ Embedding Theorem,” immediately before Theorem 4