Temporal Isometric Delay Embedding Transform

• The paper introduces TIDT, a tensorial time-series embedding that constructs an isometric Hankel representation to preserve norms and enable exact low-rank recovery.
• It unifies classical delay-embedding theory with the t-SVD framework, leveraging smoothness and periodicity to achieve robust forecasting under structured missing data.
• Empirical validations on synthetic data, network flows, urban traffic, and sea surface temperature demonstrate TIDT's superior recovery accuracy and computational efficiency.

The temporal isometric delay-embedding transform (TIDT) is a tensorial time-series embedding technique that leverages the underlying smoothness and periodicity of signals to construct an isometric Hankel (or, in the multidimensional case, Hankel-tensor) representation. TIDT enables stable, norm-preserving reconstructions from incomplete or corrupted multidimensional time series, allowing exact low-rank tensor completion and robust forecasting even under highly non-random missing data conditions. The method unifies classical delay-embedding theoretic guarantees with practical, scalable low-rank tensor recovery via the t-SVD framework. Rigorous isometry properties, explicit theoretical recovery conditions, and extensive empirical validation establish TIDT as a foundational tool for time-series analysis and multidimensional temporal data recovery (Shu et al., 11 Dec 2025).

1. Formal Definition and Isometry Properties

The core of TIDT is the construction of an isometric Hankel operator acting on a time-series. For a length-$t$ univariate series $m = (m_1, \ldots, m_t)^\top \in \mathbb{R}^t$ and a window length $k \le t$, the isometric Hankel transform is:

$$\mathcal{H}_k(m) = \frac{1}{\sqrt{k}} [m, S(m), S^2(m), \ldots, S^{k-1}(m)] \in \mathbb{R}^{t \times k}$$

where $S$ denotes the circular shift operator. The operator satisfies:

$$\|\mathcal{H}_k(m)\|_F^2 = \|m\|_2^2, \qquad \|\mathcal{H}_k(m^1) - \mathcal{H}_k(m^2)\|_F = \|m^1 - m^2\|_2$$

Thus, $\mathcal{H}_k$ is an isometry: the Frobenius norm of Hankel matrices reflects the input $\ell_2$ norm, and distances are strictly preserved. For a $p$-way tensor $$M \in \mathbb{R}^{t \times n_1 \times \cdots \times n_p}$$, TIDT generalizes by applying the same isometric Hankelization tube-wise along the temporal mode, yielding a $(p+2)$-order Hankel tensor:

$$\mathcal{H}_k(M) \in \mathbb{R}^{t \times k \times n_1 \times \cdots \times n_p}$$

This isometry extends globally:

$$\|\mathcal{H}_k(M^1) - \mathcal{H}_k(M^2)\|_F = \|M^1 - M^2\|_F$$

ensuring norm and distance preservation across the multidimensional time series (Shu et al., 11 Dec 2025).

2. Theoretical Underpinnings in Delay-Embedding Geometry

TIDT builds upon and extends the geometry-preserving delay-embedding framework originally established in deterministic dynamical systems theory. Takens' embedding theorem guarantees (under smooth generic conditions) that $m \geq 2d+1$-length delay maps reconstruct the state space topology of a $d$-dimensional attractor (Ostrow et al., 17 Jun 2024). Recent advances formulate stronger geometry-preserving statements, requiring that embeddings be bi-Lipschitz (i.e., nearly isometric within specified distortion bounds), subject to sufficient "stable rank" or "soft rank" of the induced trajectory or Hankel matrices (Yap et al., 2014, Eftekhari et al., 2016).

For Hankel-based embeddings, isometry holds when the stable rank of the delay-formed matrix/tensor scales at least with the attractor dimension, stabilizing the embedding against both redundancy (overly correlated delays) and irrelevancy (excessively separated delays) (Eftekhari et al., 2016). TIDT provides a constructive isometry by combining this insight with a particular circular-shifted (rather than strictly consecutive) Hankelization, guaranteeing norm preservation by construction (Shu et al., 11 Dec 2025).

3. Low-Rankness via Temporal Smoothness and Periodicity

A principal motivation for TIDT is the empirical and theoretical observation that real-world multivariate time series often exhibit high temporal smoothness or approximate periodicity. Key bounds formalize how these properties induce low-rank structure in the temporal Hankel representations:

• For smooth $m$, with defined $\eta(m) = \|m - S(m)\|_2$, the best rank-$r$ Hankel approximation error satisfies

$$\epsilon_r(\mathcal{H}_k(m)) \leq \sqrt{ \frac{k-r}{3k} \lceil k/r \rceil\, \eta(m) }$$

• For nearly periodic sequences (period $\tau$), $\beta_\tau(m) = \|m-N(m)\|_2$ ($N$ a $\tau$-shift):

$$\epsilon_r(\mathcal{H}_k(m)) \leq \frac{\tau}{ \sqrt{k}\, (\lceil k/\tau \rceil - 1) } \beta_\tau(m)$$

These bounds extend to the multidimensional setting: each temporal "tube" exhibits nearly low-rank Hankel structure if the corresponding sequence is smooth or periodic, and thus the full Hankel tensor inherits low tubal-rank in the t-SVD sense (Shu et al., 11 Dec 2025).

4. Tensor Completion and Exact Recovery Guarantees

TIDT enables versatile and theoretically principled tensor-completion of incomplete or corrupted time series. The associated recovery model, "Low-Rank Tensor Completion with Temporal Isometric Delay-embedding Transform" (LRTC-TIDT), minimizes the t-SVD tensor nuclear norm $\|\mathcal{H}_k(X)\|_{\circledast}$ under the constraint that the estimated $X$ matches observations on the available data mask:

$$\min_X \|\mathcal{H}_k(X)\|_{\circledast} \quad \text{s.t.} \quad P(X) = P(M)$$

where $P$ projects onto the observed entries. Under mild incoherence conditions on the t-SVD singular vectors and providing that the minimum temporal sampling rate

$$\rho(\Omega) = \min_{1 \leq i_j \leq n_j} \frac{ \#\{ i_t : (i_t, i_1, \ldots, i_p) \in \Omega \} }{t }$$

exceeds $1 - \frac{k}{2\mu r(r_s + 1)t}$ (where $r$ is the tubal-rank and $r_s$ the multi-rank sum), exact recovery of $M$ from arbitrary (possibly highly non-random) missing patterns is guaranteed. In the presence of observation noise $E$, approximate recovery with explicit error bounds is established:

$\|\hat{X} - M\|_F \leq C(\alpha, r_s) \delta$

where $\delta$ is the noise energy, $0 < \alpha < 1$ determines the sampling margin, and $C(\alpha, r_s) \to 0$ as $\alpha \to 0$ (Shu et al., 11 Dec 2025).

5. Algorithmic Implementation and Complexity

The LRTC-TIDT estimator employs an augmented Lagrangian ADMM strategy. The primary variables are $X$ and its Hankel-tensorized form $Z = \mathcal{H}_k(X)$. Alternating minimization consists of:

1. t-SVD-based singular value thresholding (t-SVT) on $Z$ in the Fourier domain across frontal slices,
2. A closed-form update of $X$ by solving a diagonal linear system leveraging the isometry of $\mathcal{H}_k$ and observation masking.

Per-iteration computational complexity is $$O(t k n_1\cdots n_p(n_1+\cdots+n_p) + t k^2 n_1 \cdots n_p)$$. The isometric property of TIDT allows for numerically stable updates and efficient implementation. Pseudocode and code for these procedures are provided in (Shu et al., 11 Dec 2025).

6. Empirical Performance and Application Domains

Empirical validation includes both synthetic and real-world scenarios with structured missingness:

• Synthetic tensors: Phase transitions in success/failure rates for recovery align precisely with predicted theoretical sampling bounds.
• Network flow reconstruction: On $204 \times 12 \times 12$ origin-destination Abilene flow data with up to 80% non-random missing entries, LRTC-TIDT obtains the lowest MAE/RMSE versus all published baselines.
• Urban traffic estimation: Daily taxi flow data ($60 \times 69 \times 69$) with noise and large block-wise missingness, where LRTC-TIDT again outperforms methods such as TNN, MDT-Tucker, and CNNM.
• Temperature field prediction: On $60 \times 30 \times 84$ monthly sea surface temperature tensors, LRTC-TIDT enables accurate forecasting with non-randomly removed future horizons, outperforming state-of-the-art competitors across all horizons.

A key property is TIDT's ability to redistribute structured missing patterns such that each lateral slice in Hankel space remains sufficiently sampled for recovery, leveraging the low tubal-rank induced by smoothness or periodicity (Shu et al., 11 Dec 2025).

7. Connections to Broader Delay-Embedding and Neural Sequence Theory

TIDT occupies a central role in the modern synthesis of delay-embedding theory, geometry-preserving embeddings, and sequence modeling in machine learning. Recent work demonstrates that the hidden states of state-space models and transformer architectures implicitly act as learned delay embeddings, whose geometry and predictive quality can be directly measured in terms of isometric properties and neighborhood preservation (Ostrow et al., 17 Jun 2024). TIDT offers an explicit analogue, providing provable guarantees of isometry and low-rankness, thus serving as a bridge between theoretical dynamical systems, tensor signal processing, and contemporary machine learning paradigms.

Isometry and low-rankness are the joint pillars enabling both classical attractor reconstruction (in the sense of Takens) and modern data recovery under non-ideal, high-dimensional scenarios. This consolidates TIDT's significance in both theoretical and practical advances in temporal data analysis.