Stable Takens' Embedding Theorem for Non-Uniformly-Sampled Linear Systems

Abstract: Takens' time-delay embedding theorem provides conditions under which delay-coordinate maps, formed using uniformly-sampled time series of trajectories evolving on attractors of dynamical systems, can faithfully represent the dynamics of the original system. Nonlinear systems can be highly sensitive, and Takens' theorem does not provide guarantees about the stability of time-delay embeddings. In the linear setting, statements about the stability of time-delay embeddings are more tractable and have been proven for delay-coordinate maps with evenly-spaced delays. In many experimental applications, however, time series data may be non-uniformly-sampled, especially in systems with multiple timescales or when using event-based rather than time-based sampling techniques. In this paper, we extend the theorems for the stable linear Takens' embeddings to the setting where the delay-coordinate maps involve unevenly-spaced delays. We pose a conjecture about the rank of generalized Vandermonde matrices that capture the temporal structure of time-delay embeddings. We prove that, provided the conjecture holds, existing theorems about stable linear Takens' embeddings readily extend to unevenly-sampled settings, and the quality of the embedding converges to the same asymptotic bounds when using a large number of delays in the delay-coordinate map, a result which is supported by numerical simulations.

• The paper extends stable linear Takens embedding theory to non-uniform delays by factoring the embedding matrix into observability and generalized Vandermonde components, with stability guaranteed conditionally on a full-rank conjecture.
• The analysis shows that random sampling schemes can achieve asymptotically optimal conditioning as the number of delays grows, while finite-delay guarantees and convergence rates remain unavailable because no determinant formula exists.
• Numerical experiments indicate that uniform sampling typically converges faster and outperforms irregular sampling for single-scale systems, suggesting that non-uniform delays are most useful for multi-scale, mobile-sensor, or event-based applications.

Overview

Takens' time-delay embedding theorem guarantees that, for generic observation functions and delays, delay-coordinate maps of trajectories on a low-dimensional attractor are embeddings. The theorem is existential and prevalence-based; it says nothing about the stability of a particular embedding for a particular choice of $h$ and $\tau$. Yap and Rozell addressed this gap for linear dynamical systems of class $A(d)$, proving stable linear Takens' embeddings with explicit conditioning bounds—but only for evenly-spaced delays. This paper by Ng and Kutz extends those stability results to delay-coordinate maps built from unevenly-spaced lags $\{\tau_k\}_{k=1}^\ell$, motivated by multi-scale dynamics, mobile sensors, and event-based (e.g., interspike-interval) sampling (2608.14001).

The central technical object is the factorization of the embedding matrix as $G = TH$, where $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$ captures the spatial observability and $T \in \mathbb{C}^{\ell \times 2d}$ is a generalized Vandermonde matrix with entries $z_p^{\tau_k}$ for nodes $z_p = e^{\pm i\theta_j} \in S^1$ and non-integer, non-consecutive powers $\tau_k$. Unlike classical Vandermonde matrices, no closed-form determinant exists for this object, so the rank question cannot be settled by the standard product formula.

A conjecture on generalized Vandermonde rank

The paper's pivotal claim is Conjecture 1: if the powers $\tau$0 are distinct positive reals and the nodes $\tau$1 have distinct angles in $\tau$2, then $\tau$3 with probability one. The authors verify the conjecture explicitly for $\tau$4, deriving aliasing conditions of the form $\tau$5 where $\tau$6 is a lag difference. Two observations from these base cases deserve emphasis:

• Non-uniform sampling can be more alias-prone than uniform sampling. When the lag difference $\tau$7 is rational ($\tau$8 in lowest terms), the aliasing set on $\tau$9 grows with $A(d)$0; when $A(d)$1 is irrational, the aliased frequency differences form a dense—though measure-zero—subset of the interval. Generic choices remain full rank, but the deterministic guarantee available for integer powers is lost.
• More lags mitigate aliasing, since only one of the $A(d)$2 row pairings must avoid the aliasing condition.

For general $A(d)$3, the determinant involves rational functions in combinatorial proliferation, and the conjecture is left open—a limitation that conditions every downstream theorem.

Determinant and condition number analysis

To analyze $A(d)$4 without a determinant formula, the authors study $A(d)$5 via Gerschgorin's circle theorem. The scaling $A(d)$6 normalizes the Frobenius "energy" per row so that $A(d)$7, making the rectangular case comparable to the square case. Three results follow:

1. Upper bound on the pseudo-transfinite diameter (their Definition 4): $A(d)$8, converging to 1 at rate $A(d)$9 independently of $\{\tau_k\}_{k=1}^\ell$0. This rules out unbounded singular values but, notably, provides no lower bound on the scaled determinant—the authors concede this asymmetry explicitly.
2. Asymptotic exactness: if $\{\tau_k\}_{k=1}^\ell$1, where $\{\tau_k\}_{k=1}^\ell$2 is the limiting Gerschgorin radius expressed through characteristic functions $\{\tau_k\}_{k=1}^\ell$3, then as $\{\tau_k\}_{k=1}^\ell$4: $\{\tau_k\}_{k=1}^\ell$5, $\{\tau_k\}_{k=1}^\ell$6 (the upper bound is sharp), and $\{\tau_k\}_{k=1}^\ell$7.
3. Finite-$\{\tau_k\}_{k=1}^\ell$8 condition number bound: if $\{\tau_k\}_{k=1}^\ell$9, then the normalized condition number satisfies $G = TH$0.

The key assumption $G = TH$1 is verified analytically for three i.i.d. sampling schemes: uniform ($G = TH$2 decays as a sinc), exponential (rate chosen so mass concentrates in $G = TH$3), and normal (variance scaling with $G = TH$4). In all three cases the characteristic function vanishes as $G = TH$5, so the asymptotic conclusions hold. Numerical experiments with $G = TH$6 frequencies, 50 trials each, confirm full rank empirically and convergence of the scaled determinant and condition number to 1, though small-$G = TH$7 deviations for exponential and normal schemes appear—attributed to numerical artifacts of severely ill-conditioned matrices rather than proven rank deficiency.

Stable embeddings with non-uniform delays

Conditioned on Conjecture 1, two main theorems extend Yap–Rozell:

• Existence (Theorem 5): for class-$G = TH$8 systems with $G = TH$9, distinct $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$0-eigenvalues, and $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$1 for all $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$2, the non-uniform delay map satisfies the bi-Lipschitz-type stability inequality with probability one for some $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$3 and $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$4. The proof reduces to $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$5 when both factors are full rank.
• Asymptotic stability (Theorem 6): as $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$6, the embedding quality converges to constants $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$7 and $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$8—the same asymptotic bounds as the uniformly-sampled case. Isometry requires $H = \diag(v_1^T h, v_1^H h, \dots, v_d^T h, v_d^H h)$9, i.e., uniform attractor geometry ($T \in \mathbb{C}^{\ell \times 2d}$0) and uniform observability ($T \in \mathbb{C}^{\ell \times 2d}$1).

A structural limitation is stated plainly: unlike the uniform case, where Gerschgorin radii reduce to geometric sums yielding an explicit finite-$T \in \mathbb{C}^{\ell \times 2d}$2 correction $T \in \mathbb{C}^{\ell \times 2d}$3, the non-geometric structure of unevenly-spaced samples prevents any closed-form enveloping bound. Consequently, no convergence rate or finite-$T \in \mathbb{C}^{\ell \times 2d}$4 guarantee is available for the non-uniform setting. A remark also extends the framework to time-dependent observation vectors $T \in \mathbb{C}^{\ell \times 2d}$5 modeling mobile sensors, with $T \in \mathbb{C}^{\ell \times 2d}$6 redefined as extrema over time.

Numerical validation

Simulations use an ambient dimension $T \in \mathbb{C}^{\ell \times 2d}$7, attractor dimension $T \in \mathbb{C}^{\ell \times 2d}$8, up to $T \in \mathbb{C}^{\ell \times 2d}$9 lags drawn from truncated uniform, exponential, and normal distributions, with 5,000 pairwise trials per $z_p^{\tau_k}$0 measuring the empirical conditioning $z_p^{\tau_k}$1. Three regimes are tested:

Case	Attractor	Observation	$z_p^{\tau_k}$2	Outcome
1	Ideal ($z_p^{\tau_k}$3)	Ideal ($z_p^{\tau_k}$4)	0	Converges toward isometry
2	Ideal	Non-ideal	0.1210	Converges to $z_p^{\tau_k}$5, not isometric
3	Non-ideal	Non-ideal	0.9684	Poor conditioning, but non-singular

In all cases, $z_p^{\tau_k}$6 and $z_p^{\tau_k}$7 converge to the predicted asymptotic bounds as $z_p^{\tau_k}$8 grows, with low variance of $z_p^{\tau_k}$9 relative to its range. Two findings stand out. First, the asymptotic limit depends only on the system coordinates and observation vector—not on the sampling scheme—so all random schemes converge to the same bound as the deterministic baseline. Second, and somewhat contrary to the motivation for non-uniform sampling, the deterministic uniformly-spaced map consistently outperforms and converges faster than every non-uniform variant tested, consistent with the stricter aliasing conditions identified in the base cases of the conjecture.

Limitations and open questions

The paper's results are conditional: both embedding theorems hold provided Conjecture 1 is true, which remains unproven for $z_p = e^{\pm i\theta_j} \in S^1$0. The determinant analysis yields only an upper bound, not the lower bound needed for unconditional stability guarantees. The absence of a $z_p = e^{\pm i\theta_j} \in S^1$1 analogue means no rates of convergence can be certified for non-uniform maps. The simulations raise, without resolving, whether uniform sampling is optimal for linear systems lacking multi-scale structure, and whether a better basis for constructing non-uniform delay maps would change the comparison. Extension to genuinely multi-scale linear systems and to multivariate observations via multiple measurement functions is deferred to future work.

Conclusion

This work extends stable linear Takens' embedding theory from evenly-spaced to unevenly-spaced delays, showing that—conditional on a plausible full-rank conjecture for generalized Vandermonde matrices—non-uniform delay-coordinate maps are stable embeddings whose asymptotic quality matches the uniform case exactly. The empirical evidence supports the conjecture across small cases, asymptotics, and simulation, while the theory candidly exposes what is missing: a proof of the conjecture for general $z_p = e^{\pm i\theta_j} \in S^1$2, lower bounds on the scaled determinant, and finite-$z_p = e^{\pm i\theta_j} \in S^1$3 conditioning estimates. The finding that uniform sampling empirically dominates non-uniform sampling for single-scale systems sharpens the practical guidance: irregular sampling should be reserved for settings—multi-scale dynamics, event-based acquisition—where it is forced or structurally advantageous.