Low-Rank Hankel Tensor Completion

• Low-Rank Hankel Tensor Completion is a framework that lifts incomplete signals into Hankel-structured tensors and recovers data using low-rank priors.
• It incorporates various models such as CP decomposition, balanced unfolding, and Tucker factorization tailored for applications like spectroscopy and traffic estimation.
• Advanced optimization methods including ADMM and scaled gradient descent ensure convergence and efficiency in large-scale, nonconvex recovery tasks.

Searching arXiv for recent and foundational papers on low-rank Hankel tensor completion.

Low-rank Hankel tensor completion denotes a family of recovery methods in which a partially observed signal, matrix, or tensor is lifted into a Hankel-structured higher-order tensor and reconstructed under a low-rank prior adapted to that lifted object. In the literature, this framework appears in at least three distinct but closely related forms: recovery of $N$-dimensional exponential signals via CP structure with Hankel-matrix nuclear-norm regularization on factor vectors; spatiotemporal traffic speed estimation via a fourth-order Hankel tensor and a truncated nuclear norm on a balanced unfolding; and multi-measurement spectral compressed sensing via a third-order Hankel tensor with low multilinear-rank Tucker structure and scaled gradient descent. Across these formulations, the central premise is that Hankelization exposes shift-invariant local structure, while completion exploits low-rankness in a model-specific sense (Ying et al., 2016, Wang et al., 2021, Li et al., 7 Jul 2025).

1. Structural model and Hankel lifting

In the exponential-signal setting, the observed tensor $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$ is modeled by

$$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$

with only a subset of entries indexed by

$$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$

available through the sampling operator $\mathcal P_\Omega$. The reconstruction target is the full tensor $\mathcal Y$, and the formulation explicitly exploits both low CP rank and exponential factor vectors (Ying et al., 2016).

In the traffic-speed setting, the starting point is an incomplete matrix $\boldsymbol Y\in\mathbb R^{N\times T}$. A two-way delay embedding

$$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$

is defined by

$$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$

thereby converting the original matrix into a fourth-order Hankel tensor $X$. Its inverse $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$0 averages all copies of each original entry. This construction is used to approximate and characterize both global and local spatiotemporal patterns in a data-driven manner (Wang et al., 2021).

In the multi-measurement spectral compressed sensing setting, $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$1 equally sampled length-$$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$2 time series are arranged as $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$3, and a third-order Hankel-tensor lifting

$$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$4

produces a block-Hankel tensor. For spectral sparse signals, the lifted tensor admits a Tucker form with

$$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$5

so Hankel lifting converts spectral sparsity into low multilinear rank (Li et al., 7 Jul 2025).

2. Rank models and optimization formulations

The phrase “low-rank” is not uniform across this literature. The formulations differ in the rank surrogate, in the object to which that surrogate is applied, and in the way the Hankel structure is enforced.

Setting	Lifted object	Rank surrogate / structure
$$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$6-dimensional exponential signals	Factor-vector Hankel matrices and CP tensor	Nuclear norm of $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$7 plus CP decomposition
Traffic speed estimation	Fourth-order Hankel tensor with balanced unfolding $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$8	Truncated nuclear norm of $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$9
Multi-measurement spectral compressed sensing	Third-order block-Hankel tensor $$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$0	Low multilinear rank $$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$1 with Tucker decomposition

For $$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$2-dimensional exponential signals, the reconstruction variable is written as

$$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$3

where $$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$4 is an overestimate of the true rank. The operator $$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$5 maps a vector into an $$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$6 Hankel matrix with $$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$7, and the model minimizes

$$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$8

This couples a CP decomposition with the nuclear norm of Hankel matrices so that low-CP-rank structure and exponential structure are enforced simultaneously (Ying et al., 2016).

For traffic speed estimation, the model solves

$$y_{i_1,\ldots,i_N}=\sum_{r=1}^{R} d_r \prod_{n=1}^{N} z_{n,r}^{\,i_n-1},$$9

Here $$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$0 is the balanced spatiotemporal unfolding of the fourth-order Hankel tensor, with

$$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$1

and the truncated nuclear norm is used to approximate the tensor rank. Each column of the unfolding represents the vectorization of a small patch in the original matrix (Wang et al., 2021).

For multi-measurement spectral compressed sensing, the constrained completion problem is

$$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$2

subject to

$$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$3

Equivalently, one may add a quadratic penalty for $$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$4. The weighting operator $$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$5 is chosen so that $$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$6 satisfies $$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$7, and the Hankel tensor is parametrized by a Tucker decomposition $$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$8 (Li et al., 7 Jul 2025).

3. Numerical algorithms and computational structure

The 2016 HMRTC method solves a nonconvex and nonsmooth objective by ADMM after introducing auxiliary variables $$\Omega\subset\{1\ldots I_1\}\times\{1\ldots I_2\}\times\cdots\times\{1\ldots I_N\}$$9. The augmented Lagrangian uses multipliers $\mathcal P_\Omega$0 and penalty $\mathcal P_\Omega$1. The $\mathcal P_\Omega$2-update fixes $\mathcal P_\Omega$3 and $\mathcal P_\Omega$4, then minimizes the augmented Lagrangian; because the CP term is multilinear, an inner alternating minimization among the modes is applied, and the mode-$\mathcal P_\Omega$5 update reduces to a regularized least-squares system. The $\mathcal P_\Omega$6-update is a singular-value-shrinkage step,

$\mathcal P_\Omega$7

followed by the multiplier update

$\mathcal P_\Omega$8

The penalty schedule starts at $\mathcal P_\Omega$9 and increases by $\mathcal Y$0 with $\mathcal Y$1, for example $\mathcal Y$2. The stopping criteria are relative change in reconstructed tensor $\mathcal Y$3 or $\mathcal Y$4. Per iteration, SVDs dominate the cost: one SVD of size $\mathcal Y$5 per $\mathcal Y$6, i.e. $\mathcal Y$7, for total $\mathcal Y$8; these SVDs are independent and can be parallelized, and storage is $\mathcal Y$9 (Ying et al., 2016).

The traffic-speed STH-LRTC framework also uses ADMM, but the variables are the lifted Hankel tensor $\boldsymbol Y\in\mathbb R^{N\times T}$0, the completed matrix $\boldsymbol Y\in\mathbb R^{N\times T}$1, and a dual variable $\boldsymbol Y\in\mathbb R^{N\times T}$2. The $\boldsymbol Y\in\mathbb R^{N\times T}$3-update applies the truncated singular-value shrinkage operator $\boldsymbol Y\in\mathbb R^{N\times T}$4 to the balanced unfolding $\boldsymbol Y\in\mathbb R^{N\times T}$5, where

$\boldsymbol Y\in\mathbb R^{N\times T}$6

The $\boldsymbol Y\in\mathbb R^{N\times T}$7-update uses $\boldsymbol Y\in\mathbb R^{N\times T}$8 together with a hard constraint on $\boldsymbol Y\in\mathbb R^{N\times T}$9, the dual update is

$$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$0

and the adaptive penalty rule is

$$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$1

The stopping criterion is $$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$2. Each $$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$3-update requires an SVD of a $$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$4 unfolding with $$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$5, so a partial SVD costs $$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$6, while the $$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$7- and $$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$8-updates cost only $$\mathcal H_{\tau_s,\tau_t}:\mathbb R^{N\times T}\longrightarrow \mathbb R^{\tau_s\times\tau_t\times (N-\tau_s+1)\times (T-\tau_t+1)}$$9 (Wang et al., 2021).

The 2025 ScalHT method replaces ADMM by scaled gradient descent on Tucker factors and core. At iterate $$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$0, each factor update is premultiplied by the inverse Gram matrix of the other factors, and the core update is scaled by $$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$1, $$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$2, and $$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$3. The paper states that this scaling dramatically reduces sensitivity to condition-number in practice. Fast identities exploiting the interaction between the low-rank Tucker structure and the Hankel lift reduce the overall per-iteration cost to

$$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$4

with FFT-based fast convolution for single-mode multiplications. Initialization is “sequential spectral,” based on a back-projected estimate on a small batch $$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$5, top-$$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$6 singular vectors of mode unfoldings, a projected core $$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$7, and projection of $$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$8 to the incoherence ball (Li et al., 7 Jul 2025).

4. Convergence theory and recovery guarantees

Theoretical guarantees differ sharply across formulations. In HMRTC, despite nonconvexity, two convergence statements are established: Theorem 1 states that the sequence $$\bigl[\mathcal H_{\tau_s,\tau_t}(Z)\bigr]_{a,b,i,j}=Z_{i+a-1,\;j+b-1},$$9 generated by the ADMM is Cauchy, hence convergent; Theorem 2 states that if the multiplier increments $X$0, then any limit point satisfies the Karush–Kuhn–Tucker conditions of the original problem. No global exact-recovery or sample-complexity bounds are given for the $X$1-dimensional case, although the paper refers to existing $X$2-D Hankel-matrix completion results predicting sample complexity $X$3 and to EMaC results (Ying et al., 2016).

For STH-LRTC, the theory reported is more generic: ADMM is known to converge, under mild conditions, at roughly a sublinear rate, and in practice an adaptive $X$4 often accelerates convergence. The principal methodological parameters are the spatial and temporal window lengths $X$5, which control the size of local patches. Larger values capture longer spatiotemporal dependencies, which is useful when data is very sparse, at the expense of higher SVD cost (Wang et al., 2021).

ScalHT provides the strongest formal guarantees among the three formulations. Under an incoherence assumption on the HOSVD factors and with step-size $X$6, Theorem 1 states that if

$X$7

then, with probability $X$8, the iterates satisfy

$X$9

The paper further states exact recovery in $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$00 steps. Lemma 7 gives linear convergence once the iterate lies within a small constant relative neighborhood of the ground truth: $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$01 The proof relies on a new concentration bound for partial sampling through the Hankel adjoint $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$02, a sequential spectral initialization bound, and perturbation bounds for the Gram scaling. The authors describe these recovery and linear convergence guarantees as the first of their kind for low-rank Hankel tensor completion (Li et al., 7 Jul 2025).

5. Empirical behavior and application domains

The empirical record in the 2016 paper is centered on simulated and real spectroscopy data. On simulated $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$03 three-dimensional signals, including both undamped and damped complex sinusoids, with rank $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$04 varying up to $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$05, sampling ratios $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$06 from $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$07 to $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$08, and additive noise, HMRTC yields much lower RLNE than competing low-n-rank (ADM–TR) or weighted-CP (WCP) methods, especially at low $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$09. Phase-transition plots of average RLNE show that HMRTC recovers accurately, with RLNE below the noise level, with as few as $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$10 for moderate $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$11, whereas the competing methods need $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$12. HMRTC is also reported to be robust to over-estimation of rank: even if $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$13 is $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$14, reconstruction remains good, whereas WCP fails when $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$15. On real three-dimensional NMR spectroscopy (HNCO) data of size $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$16, nonuniformly Poisson-gap sampled at $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$17, HMRTC recovers peaks faithfully versus ADM–TR or WCP, enabling $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$18 time-saving. In a $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$19 benchmark, HMRTC runs in $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$20 s versus $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$21 s for WCP and $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$22 s for ADM–TR, using $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$23 GB RAM (Ying et al., 2016).

The 2021 traffic-speed formulation is motivated by traffic state estimation using sparse observations from mobile sensors. The paper contrasts its approach with methods that rely on well-defined physical traffic flow models or require large amounts of simulation data to train machine learning models. The proposed framework is purely data-driven and model-free, involves only two hyperparameters, spatial and temporal window lengths, and numerical experiments on real-world high-resolution trajectory data demonstrate the effectiveness and superiority of the model in some challenging scenarios (Wang et al., 2021).

The 2025 ScalHT paper reports several numerical regimes. In phase-transition experiments with $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$24, $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$25, and $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$26, success occurs roughly when $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$27; with $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$28 fixed, $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$29 suffices. For $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$30, $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$31, $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$32, ScalHT, ScaledGD, and AM-FIHT achieve high success at $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$33, while ANM needs $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$34. For $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$35, $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$36, $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$37, ScalHT and ScaledGD reach relative error $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$38 in $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$39 iterations in both $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$40 and $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$41 cases, whereas AM-FIHT takes $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$42 to $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$43 iterations. Across $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$44, ScalHT runs in $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$45 time and is reported at $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$46 sec for $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$47. In DOA estimation using a $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$48-element sparse linear array with $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$49, ScalHT+MUSIC achieves RMSE near the CRB at SNR $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$50 dB, outperforming ANM+MUSIC and applying MUSIC directly to incomplete snapshots (Li et al., 7 Jul 2025).

6. Conceptual distinctions, adjacent methods, and interpretation

A recurrent source of confusion is to treat low-rank Hankel tensor completion as a single model. The literature instead shows several structurally different formulations. One line exploits the CANDECOMP/PARAFAC structure together with the exponential structure of factor vectors by minimizing the nuclear norm of their Hankel matrices; another uses a balanced unfolding of a fourth-order spatiotemporal Hankel tensor and a truncated nuclear norm; a third imposes low multilinear rank on a third-order block-Hankel tensor through Tucker factors and a Hankel-consistency constraint or penalty (Ying et al., 2016, Wang et al., 2021, Li et al., 7 Jul 2025).

The relation to adjacent methods is also explicit in the comparative baselines. In the spectroscopy setting, HMRTC is compared with ADM–TR and WCP. In traffic state estimation, the proposed method is positioned against physical traffic flow models and machine learning models trained on large amounts of simulation data. In multi-measurement spectral compressed sensing, ScalHT is compared with ANM, AM-FIHT, and ScaledGD. These comparisons indicate that the practical meaning of “Hankel tensor completion” depends on the ambient inverse problem, the lifted tensor order, the rank notion, and the optimization mechanism.

The theoretical trajectory across the cited works is equally nonuniform. Early HMRTC establishes convergence to a stationary point but does not provide global exact-recovery or $$\mathcal Y\in\mathbb C^{I_1\times I_2\times\cdots\times I_N}$$51-dimensional sample-complexity results. STH-LRTC emphasizes tractable ADMM updates and patch-based spatiotemporal structure. ScalHT combines fast algebraic transforms with nonconvex Tucker factorization and provides recovery and linear convergence guarantees. This suggests a progression from convex or convex-surrogate regularization toward structured nonconvex parameterizations with sharper theory and lower computational cost, while preserving the defining role of Hankel lifting as the mechanism that exposes recoverable low-rank structure.