Tensor Completion: Models and Methods

• Tensor Completion is the process of recovering missing data in multi-dimensional arrays by leveraging low-rank structures and auxiliary relationships.
• It employs various rank models—such as CP, Tucker, TT, and t-SVD—and integrates convex optimization, manifold methods, and adaptive sampling.
• Practical applications span visual inpainting, hyperspectral imaging, and scientific simulations, emphasizing efficient recovery and reliable performance.

Tensor completion is the problem of recovering missing entries of a partially observed multiway array by exploiting low-rank structure across modes, and, in more specialized formulations, auxiliary relational, geometric, or statistical structure. In its most general form, the observed tensor is indexed by an observation set $\Omega$, a projection operator $P_\Omega$ keeps entries in $\Omega$ and zeros out the rest, and reconstruction is posed either as exact consistency on $\Omega$ or as least-squares fitting over $\Omega$. Contemporary formulations span CP, Tucker, TT, TR, and t-SVD/tubal-rank models; convex surrogates such as sums of nuclear norms and transformed tensor nuclear norms; Riemannian optimization on low-rank manifolds; adaptive sampling; plug-and-play regularization; and deterministic completion under structured observation patterns (Song et al., 2017, Chen et al., 14 Oct 2025).

1. Problem formulation and observation models

A standard formulation starts from an $N$th-order tensor $$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$ or $$\mathcal{Y} \in \mathbb{R}^{n_1 \times \cdots \times n_d}$$, together with an observation set $\Omega \subseteq [I_1]\times\cdots\times[I_N]$. The sampling operator is defined entrywise by

$$\bigl[P_\Omega(X)\bigr]_{i_1,\dots,i_N}= \begin{cases} X_{i_1,\dots,i_N}, & (i_1,\dots,i_N)\in\Omega,\ 0, & \text{otherwise}, \end{cases}$$

and the canonical data-fitting loss is

$P_\Omega$0

Equivalent formulations appear across TT, Tucker, t-SVD, and convex latent-norm settings (Yu et al., 2019, Kapushev et al., 2019).

In the noise-free setting, one enforces exact agreement on observed entries,

$P_\Omega$1

whereas noisy formulations use a least-squares fidelity term, sometimes augmented by structural regularizers such as nuclear norms, sparse penalties, total variation, or learned denoisers (Song et al., 2017, Chen et al., 14 Oct 2025). Error measures vary with the application. Representative examples include relative Frobenius error,

$P_\Omega$2

RMSE on observed entries, PSNR and SSIM for imaging, and NRE for sparse tensor benchmarks (Kapushev et al., 2019, Ahn et al., 26 Mar 2025).

Although entrywise random sampling is the dominant model in the literature, tensor completion is not limited to it. One line of work studies fiberwise observation along a single mode: all entries in some mode-$P_\Omega$3 fibers are observed, while the remaining fibers are entirely missing. After permutation to $P_\Omega$4, the mode-$P_\Omega$5 unfolding is observed on a subset of rows, and completion becomes an algebraic reconstruction problem driven by structured submatrices rather than isolated entries (Sofi et al., 16 Sep 2025). Another line treats general linear measurements via a sensing map $P_\Omega$6, leading to tensor sensing rather than pure entrywise completion, but with closely related low-rank recovery mechanisms (Xu, 2020).

2. Rank models and structural representations

The central modeling choice is the notion of tensor rank. In the CP model,

$P_\Omega$7

and the CP rank is the minimum such $P_\Omega$8. In the Tucker model,

$P_\Omega$9

and the multilinear rank is the tuple of ranks of the mode-$\Omega$0 unfoldings (Song et al., 2017). These two models dominate classical tensor completion, but they are not the only ones.

For high-order tensors, tensor train decompositions replace a single core by a chain of 3-way cores. In TT format,

$\Omega$1

with cores $\Omega$2 and $\Omega$3. The storage scales as $\Omega$4 when $\Omega$5 and $\Omega$6, which is a principal reason TT is favored in very high-dimensional regimes (Kapushev et al., 2019). TT completion also admits a quotient-manifold formulation on the smooth manifold of fixed TT-rank tensors, with dimension

$\Omega$7

under the stated non-emptiness conditions (Cai et al., 2022).

A different algebra is used by transformed t-SVD methods. For a third-order tensor, a unitary transform $\Omega$8 along the third mode induces transformed frontal slices $\Omega$9, a transformed t-product, and a transformed t-SVD

$\Omega$0

This yields both the tubal rank,

$\Omega$1

and the multi-rank vector $\Omega$2, where $\Omega$3 (Song et al., 2020). More recent transform-based work removes the restriction to orthogonal square transforms and studies arbitrary linear transforms, including rectangular tall “slim transforms,” directly in the transform domain (Chen et al., 2024).

Geometric formulations reinterpret low-rank sets as manifolds. For CP rank-one tensors, the relevant manifold is the Segre manifold, the set of rank-1 tensors, parameterized as a cone over a product of spheres: $\Omega$4 This supports explicit geodesics and Riemannian conjugate-gradient algorithms for CP completion (Swijsen et al., 2021).

3. Convex, transform-based, and operator-theoretic formulations

One major family replaces rank minimization by convex surrogates. The most classical is the sum of nuclear norms of unfoldings,

$\Omega$5

or its noisy least-squares variant (Song et al., 2017). These methods are effective but can be expensive because they repeatedly compute large SVDs on matricizations.

A key criticism of classical latent nuclear norms is that “one mode versus the rest” matricizations become highly rectangular in high order, so the attainable rank is too small to capture global dependencies. To address this, balanced unfoldings based on tensor circular unfolding were introduced. The corresponding latent norm is

$\Omega$6

and is optimized by a Frank–Wolfe algorithm that needs only rank-one SVDs of balanced unfoldings and accesses only observed entries. The method maintains sparse tensors on $\Omega$7 and compact basis matrices rather than materializing the full tensor, and was reported to achieve state-of-the-art performance in visual-data inpainting with markedly reduced time and memory (Yu et al., 2019).

Transform-based tensor nuclear norms generalize the t-SVD approach. In the transformed setting, the convex surrogate is

$\Omega$8

and completion is posed as

$\Omega$9

The main theoretical novelty of the multi-rank formulation is that recovery bounds scale with $\Omega$0 rather than $\Omega$1, which can be substantially tighter when the transformed slice ranks are unevenly distributed (Song et al., 2020). Related work on arbitrary linear transforms studies

$\Omega$2

where $\Omega$3, and proves exact recovery with non-orthogonal and slim transforms (Chen et al., 2024).

Recent plug-and-play formulations replace an explicit local regularizer by a deep denoiser, but the operator-theoretic treatment is explicit that assuming a deep denoiser is a proximal operator of an implicit regularizer is generally invalid. The monotone-inclusion formulation

$\Omega$4

with $\Omega$5, $\Omega$6, and $\Omega$7, leads to a Davis–Yin splitting algorithm and a global convergence theorem under pseudo-contractive denoiser assumptions (Chen et al., 14 Oct 2025). This makes tensor completion one of the settings in which nonproximal learned priors have been integrated with explicit convergence theory rather than only empirical fixed-point arguments.

A different convex direction uses a gauge norm whose unit ball is the convex hull of rank-1 sign tensors. The completion problem is then solved by Blended Conditional Gradients with weak separation oracles implemented by mixed-integer linear optimization, and the statistical analysis is tied to information-theoretic rates for bounded CP-rank classes (Ge et al., 2024). This suggests that, beyond spectral surrogates, tensor completion also admits combinatorial convex formulations.

4. Factorization, manifolds, adaptive sampling, and structured initialization

Nonconvex factorization remains the dominant computational paradigm because it scales to large tensors and can exploit specific rank formats directly. In TT completion by alternating minimization, each core slice is updated by solving a masked least-squares problem while holding all other TT cores fixed. The resulting TCAM-TT algorithm works directly with left and right TT environments, and on the synthetic $\Omega$8 benchmark with TT-rank $\Omega$9 it achieved “perfect” recovery for observation ratio $N$0 (Wang et al., 2016).

Alternating minimization has also been revisited theoretically in CP completion. Kronecker Alternating Minimization updates against the full Kronecker subspace $N$1 rather than the matched-column Khatri–Rao product, which permits a contraction analysis based on principal angles of subspaces. The algorithm converges linearly under the stated incoherence and robust linear-independence assumptions, and exact completion is obtained in nearly linear time with sample complexity scaling like $N$2 up to polynomial factors in the parameters (Liu et al., 2020). This stands in explicit contrast to standard ALS, which that work shows can converge at a drastically slower rate in practice.

Manifold-based methods treat low-rank tensors as smooth quotient or embedded manifolds. For CP models, explicit geodesics on the Segre manifold serve as exact retractions in a Riemannian conjugate-gradient method. For TT models, a preconditioned quotient metric yields Riemannian gradient descent, conjugate gradient, and Gauss–Newton methods; the Gauss–Newton method on the quotient geometry is shown to be equivalent to the embedded-geometry Gauss–Newton method with a specific retraction (Swijsen et al., 2021, Cai et al., 2022). These formulations are technically important because they separate gauge degrees of freedom from the actual tangent geometry of the low-rank tensor manifold.

Another factorized approach wraps a general tensor network inside orthonormal Tucker factors. The model

$N$3

uses a tensor network representation for the core $N$4, and solves completion as a system of nonlinear equations by a two-level alternating least-squares scheme. Under the stated assumptions, the method converges to the exact solution at a linear rate with high probability (Cai et al., 2020).

Initialization has itself become an algorithmic topic. In high-order TT completion, if tensor values are generated by an unknown smooth function on a grid, a Gaussian process can be trained on observed entries and then queried by TT-cross to construct an initial TT approximation. The pipeline is: train a GP on rescaled indices, use the GP posterior mean as a black-box evaluator inside TT-cross, adaptively determine TT ranks by TT-rounding, and then initialize a downstream TT optimizer such as SGD, Ropt, TTWopt, or ALS (Kapushev et al., 2019). The empirical conclusion is that GP-based initialization improves both convergence speed and final error compared with random cores, especially for smoother generators, though ALS can overfit if the initial fit is already extremely strong (Kapushev et al., 2019).

Recent methods also use adaptive sampling rather than passive observation. “Tensor Sandwich” samples a small number of outer slices, completes them by adaptive matrix completion, runs a modified Jennrich step to recover CP factors, and then samples informative fibers selected by pivoted QR on $N$5 to solve for the remaining factor $N$6. Under the stated assumptions, it completes an $N$7 CP-rank $N$8 tensor with high probability using at most $N$9 adaptively chosen samples (Haselby et al., 2023). At the opposite extreme, when observations are structured as full fibers along one mode, TT completion can be performed by standard linear algebra alone, using subspace-intersection and least-squares recovery of TT cores under deterministic conditions on isorank submatrices and overlaps (Sofi et al., 16 Sep 2025).

Finally, not all recent factor models are purely low-rank. TGL couples CP decomposition with per-mode graph neural networks: KNN graphs are built from factor rows using cosine similarity, self-looped and symmetrically normalized, and then refined factors $$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$0 are used in the CP reconstruction

$$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$1

The paper reports that TGL outperforms standard CP decomposition and is second-best overall to CoSTCo on its reported datasets (Ahn et al., 26 Mar 2025).

5. Recovery theory, identifiability, and deterministic structure

Recovery guarantees in tensor completion are highly model-dependent. The survey literature already emphasized that matrix-style results do not transfer mechanically: sample complexity depends on the chosen rank notion, the sampling model, and incoherence or analogous spread conditions (Song et al., 2017). Later work sharpened these distinctions.

The following guarantees are representative.

Setting	Representative guarantee	Citation
Transformed t-SVD, exact completion	$$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$2 for unique TTNN recovery with high probability	(Song et al., 2020)
Arbitrary linear transforms	Exact recovery if $$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$3	(Chen et al., 2024)
CP via adaptive sampling	At most $$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$4 adaptively chosen samples with probability at least $$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$5	(Haselby et al., 2023)
CP via KAM	Exact completion in nearly linear time under the stated incoherence and conditioning assumptions, with observation rate proportional to $$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$6 up to polynomial factors	(Liu et al., 2020)
Unit-consistent completion	Existence and uniqueness under canonical constraints and full support; $$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$7 term-completion after linear-time preprocessing for fixed $$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$8	(Nguyen et al., 2022)

A major technical divide is between probabilistic and deterministic guarantees. Probabilistic results dominate entrywise sampling: they assume Bernoulli or uniform random observations together with incoherence conditions on factor subspaces or transformed singular spaces (Song et al., 2020, Liu et al., 2020). Deterministic guarantees emerge when the observation pattern itself has exploitable structure. Fiberwise TT completion along a single mode shows that if an informationally complete set of fully observed isorank submatrices exists, and the associated overlap and rank conditions hold, then the TT decomposition is recoverable by ordinary SVD, QR, and least squares (Sofi et al., 16 Sep 2025). The unit-consistent framework is deterministic in a stronger sense: rather than optimizing a surrogate rank objective, it imposes canonical unit-product constraints, proves existence of a canonical scaling solution, and proves uniqueness of completion under full support (Nguyen et al., 2022).

Several controversies in the literature are now better understood. One is that unbalanced matricizations can be too weak as global surrogates in high-order tensors, which motivated balanced circular unfoldings (Yu et al., 2019). Another is that plug-and-play denoisers should not be treated as proximal maps unless this is actually justified; the monotone-inclusion approach was developed precisely to avoid that assumption (Chen et al., 14 Oct 2025). A third is that straightforward ALS intuition from matrices may fail for tensors because the relevant progress geometry can depend on Kronecker subspaces rather than the Khatri–Rao structure used by standard updates (Liu et al., 2020).

This suggests that identifiability in tensor completion is not a single theorem but a family of model-specific statements. Rank format, transform domain, observation pattern, and geometric parameterization all materially affect what can be proved.

6. Applications, empirical behavior, and practical limitations

Tensor completion has been evaluated on visual inpainting, hyperspectral imaging, videos, recommender systems, traffic tensors, PDE surrogates, chemometric spectroscopy, circuit simulation, and weather data (Song et al., 2017). The application domain often determines which rank model is technically appropriate.

Visual data remain a central benchmark. Balanced latent-norm Frank–Wolfe methods were reported to achieve state-of-the-art performance on MRI, HSI, train-video, and ORL datasets while using smaller time and memory than ADMM- or BCD-based norm methods (Yu et al., 2019). GTCTV-DPC was reported to outperform strong baselines on CAVE hyperspectral images, YUV videos, and traffic tensors, with especially strong gains at low sampling rates (Chen et al., 14 Oct 2025). Transform-domain exact completion with slim transforms showed empirically that slim and concatenated transforms such as DFT-DCT and framelets outperform square DFT or DCT counterparts in multispectral image and video inpainting (Chen et al., 2024).

High-dimensional scientific and engineering tensors expose different tradeoffs. GP-initialized TT completion was evaluated on synthetic functions drawn from exponential, Matérn $$\mathcal{X} \in \mathbb{R}^{I_1 \times \cdots \times I_N}$$9, Matérn $$\mathcal{Y} \in \mathbb{R}^{n_1 \times \cdots \times n_d}$$0, and RBF kernels, on the “cookie problem” for a parameter-dependent PDE, and on a 57-parameter CMOS ring oscillator with ambient size $$\mathcal{Y} \in \mathbb{R}^{n_1 \times \cdots \times n_d}$$1. Across optimizers, the GP-based initialization reduced test relative errors substantially versus random initialization; smoother generating processes benefited most, while ALS could fit training data to near machine precision yet generalize poorly (Kapushev et al., 2019). Low-rank and sparse enhanced Tucker decomposition was evaluated on internet traffic tensors, real traffic speed data, color images, and face datasets, and was reported to be particularly effective for structured missing slices because its Toeplitz operators encode periodicity and smoothness (Pan et al., 2020).

Structured learning beyond low-rankness is increasingly common. TGL augments CP factors with per-mode GNN smoothing on factor-derived KNN graphs, and the reported experiments on Yelp and BBC-News tensors showed lower NRE than standard CPD but weaker results than CoSTCo; the paper explicitly notes a possible redundancy because graphs are derived from the same factors they are used to refine (Ahn et al., 26 Mar 2025). This is a useful corrective to the assumption that graph augmentation is automatically beneficial.

Several practical limitations recur across methods. Smooth-surrogate initializers such as GP-based TT-cross can be biased when the true generative function is non-smooth or discontinuous, and GP hyperparameters can be difficult to learn under extreme sparsity (Kapushev et al., 2019). t-SVD and transform-domain methods can be expensive because they require repeated SVDs across many frontal slices or ADMM inner loops (Chen et al., 14 Oct 2025, Xu, 2020). Factorized methods are nonconvex and remain sensitive to rank specification, even when rank-adaptive heuristics are available (Wang et al., 2016, Cai et al., 2022). Deterministic fiberwise completion is fast, but only under a very particular observation model and explicit overlap conditions (Sofi et al., 16 Sep 2025). Unit-consistent completion provides fairness and consensus-order guarantees, but it relies on positivity and full support rather than on a low-rank latent-variable model (Nguyen et al., 2022).

A persistent misconception is that tensor completion is merely matrix completion applied to unfoldings. The accumulated evidence argues otherwise. TT, t-SVD, Segre, tensor-network, and operator-splitting formulations all exploit genuinely multiway structure that is lost under naive matricization (Song et al., 2017, Swijsen et al., 2021). A plausible implication is that the field’s progress has come less from a single universal model than from matching the observation geometry and application structure to the appropriate tensor representation.