Low Rank Tensor Completion via Adaptive ADMM Niclas Führling , Getuar Rexhepi , and Giuseppe Thadeu Freitas de Abreu
arXiv:2605.03736v1 [stat.ML] 5 May 2026
School of Computer Science and Engineering, Constructor University, Bremen, Germany {nfuehrling,grexhepi,gabreu}@constructor.university Abstract—We consider a novel algorithm, for the completion of partially observed low-rank tensors, as a generalization of matrix completion. The proposed low-rank tensor completion (TC) method builds on the conventional nuclear norm (NN) minimization-based low-rank TC paradigm, by leveraging the alternating direction method of multipliers (ADMM) optimization framework. To that extend the original NN minimization problem is reformulated into multiple subproblems, which are then solved iteratively via closed-form proximal operators, making use of over-relaxation and an adaptive penalty parameter update scheme, to further speed up convergence and improve the overall performance of the method. Simulation results demonstrate the superior performance of the new method in terms of normalized mean square error (NMSE), compared to the conventional state-of-the-art (SotA) techniques, including NN minimization approaches, as well as a mixture of the latter with a matrix factorization approach, while its convergence can be significantly improved by initializing the algorithm with the solution of the SotA.
I. I NTRODUCTION
D
UE to the rise of machine learning and big data, matrix completion (MC) has emerged as a fundamental problem in a wide range of modern applications, that span from localization algorithms in signal processing [1], to collaborative filtering and recommendation systems [2], to more recent applications such as millimeter-wave channel estimation in wireless communications [3], and many more. Conventional MC techniques are based on a structured lowrank problem formulation [4]–[6], with the goal a low-rank matrix X ∈ Rm×n from a partially observed and incomplete matrix O ∈ Rm×n [7]–[9]. However, the increasing use of multidimensional data in various domains has highlighted the limitations of matrix completion, which has motivated the extension to tensor completion (TC). Similar to matrix completion, tensor completion aims to recover a low-rank tensor T ∈ RI1 ×I2 ×···×IN , from a partially observed tensor O of the same dimensions, under the assumption that the true tensor admits a low-dimensional latent structure [10]–[12]. Unlike matrices, however, tensors admit multiple notions of rank, such as CP rank, Tucker rank, and tubal rank, each capturing different types of multilinear dependencies. This fundamental difference makes tensor completion (TC) both more expressive and significantly more challenging than its matrix counterpart. Early TC approaches have largely been inspired by developments in MC, where the non-convex tensor rank objectives are replaced with convex or tractable surrogates. Common techniques rely on minimizing sums of nuclear norms of moden unfoldings, effectively extending nuclear norm (NN)-based
MC techniques to the tensor setting, as proposed in [13]. While such relaxations preserve convexity and theoretical guarantees, they often suffer from high computational complexity and may fail to fully exploit the intrinsic multilinear structure of the tensor. As an alternative, factorization-based formulations, which parameterize the tensor through low-dimensional latent factors, have been widely adopted due to their favorable scalability and reduced memory footprint [14]. In addition, hybrid methods that combine nuclear-norm regularization with factorized representations have been proposed to bridge the gap between convex relaxations and scalable low-rank modeling [15]. While more recent advances have proposed discrete-aware techniques for both matrix and tensor completion [16]–[18], these methods, while having shown superior performance compared to their continuous counterparts, are only applicable in limited applications. In light of the above, we propose an extension of the tensor completion (TC) paradigm via a novel low-rank TC method, that leverages alternating direction method of multipliers (ADMM) to solve the NN minimization problem, by reformulating the original problem into multiple subproblems. These subproblems can then be solved iteratively in closedform, also leveraging over-relaxation and an adaptive penalty parameter update scheme. The rest of the article is structured as follows. First, basics on the TC problem, as a generalization of the MC problem, with a description of conventional TC techniques is given in Section II. Then, the proposed alternating direction method of multipliers (ADMM) method is described in Section III, followed by numerical results that compare our contribution with state-of-the-art (SotA) methods and a few concluding remarks, in Sections IV and V, respectively. II. P RELIMINARIES In general, tensors are a multi-dimensional generalization of vectors and matrices, such that an N-mode tensor T can be written as T ∈ RI1 ×I2 ×···×IN , (1) where In denotes the size of the tensor in its n-th mode. For the sake of simplicity, we perform the tensor completion in terms of image processing, where a partially observed, sampled version of an image is received, which needs to be recovered. A. Conventional Tensor Completion Problem Formulation Conventional SotA low rank TC techniques are based on a minimizing the rank of the tensor. However, to introduce the
concept, the problem can be simplified to a MC problem, such that the original rank minimization can be written as (2a) arg min rank(X), X∈Rm×n
PΩ (X) = PΩ (O),
s.t.
(2b)
where rank(·) denotes the rank operator of the input matrix, and PΩ (·) indicates the mask ( operator, which is defined as [X]i,j , if (i, j) ∈ Ω, (3) [PΩ (X)]i,j = 0, otherwise, where Ω defines the index set that contains the indices of the observed elements and [·]i,j denotes the (i, j)-th element of a given matrix. While problem (2) is optimal, it has been shown in [19] that the problem is non-deterministic polynomial-time (NP)hard due to the non-convexity of the rank operator, such that its solution cannot be obtained efficiently, which applies to both, matrices and tensors. However, it has also been shown in [20] that the rank minimization problem can be relaxed by replacing the rank objective with the NN operator ||X||∗ , which is given by the sum of the singular values of X, since a matrix of rank r has exactly r nonzero singular values. The relaxed optimization problem utilizing the NN objective can be therefore written as arg min
||X||∗ ,
(4a)
PΩ (X) = PΩ (O),
(4b)
X∈Rm×n
s.t.
which can be viewed as a tight lower bound of the rank operator [21], compared to the original problem in (2). Even though enforcing constraint (4b) exactly is technically correct, in practical scenarios, the observed matrix typically contains measurement noise. To that extent, the low-rank requirement can be relaxed, such that the target matrix is recovered as an approximately low-rank solution. In this context, several related works, e.g., [22]–[24], have considered the following relaxed formulation of problem (4), defined as arg min
||X||∗ ,
(5a)
1 ||PΩ (X − O)||2F ≤ ϵ, 2 | {z }
(5b)
X∈Rm×n
s.t.
≜f (X)
which can be rewritten in regularized form as arg min
f (X) + λ||X||∗ .
(6)
where X(n) denotes P the mode-n matricization of the tensor X and αn ≥ 0, αn = 1, which can be further rewritten as arg min X
s.t
s.t
PΩ (X ) = PΩ (T ),
where the nuclear norm for a Tensor is defined as N X ||X ||∗ = αn ||X(n) ||∗ n=1
(7b)
(8)
(9a)
PΩ (X ) = PΩ (T ).
(9b)
To elaborate, mode-n matrizitation rearranges the elements of a tensor into a two-dimensional (2D) matrix. For a tensor T ∈ RI1 ×I2 ×···×IN , the mode-n unfolding is a matrix, defined as Q T(n) ∈ RIn × j̸=n Ij . (10) B. Soft-impute Approach A straightforward approach to solve the NN minimization problem in (6) is to use the soft-impute (SI) method, which is not only relevant in the context of matrix completion, but also in the context of tensor completion, since the same concept can be applied to the mode-n matricization of the tensor, as discussed above. To be more specific, SI [25] and its extension the accelerated and inexact SI (AIS)-impute approach of [26], are recently proposed methods that solve large-scale MC problems [27], [28], which have a similar structure as the problems described by equations (5) and (6) that aim to solve the NN minimization problem efficiently by using singular value thresholding (SVT). In the conventional SI approach the following recursion is used, given by Xt = SVTλ (Xt−1 + PΩ (O − Xt−1 )),
(11)
where the SVT function is defined as SVTλ (A) = U (Σ − λI)+ V ⊺ ,
(12)
in which (·)+ defines the positive part of the input and A ≜ U ΣV ⊺ . III. P ROPOSED T ENSOR C OMPLETION M ETHOD A. Problem Formulation Let T ∈ RI1 ×I2 ×I3 be a partially observed low-rank tensor, with the set of observed indices denoted as Ω and the corresponding masking operator PΩ (·). Revisiting the problem in (7), we seek to recover T by solving the Sum-of-NuclearNorms minimization problem, written as N X λ αn X(n) ∗ , arg min (13a) X ∈RI1 ×I2 ×I3
s.t.
X
αn ||X(n) ||∗ ,
n=1
X∈Rm×n
Next, as mentioned before, the same concept of nuclear norm minimization can be applied to Tensors. For X , T = RI1 ,I1 ,···IN being N-mode Tensors the problem can be rewritten as arg min ||X ||∗ , (7a)
N X
Q
n=1
PΩ (X ) = PΩ (T ),
(13b)
where X(n) ∈ RIn × j̸=n Ij is the mode-nP unfolding of X , αn ≥ 0 are mode-wise weights satisfying n αn = 1, and λ > 0 controls the degree of nuclear-norm regularization. Problem (13) was originally addressed by the algorithm in [13], which solve it via ADMM with a fixed penalty parameter. While convergent, fixed-penalty ADMM converges at a rate O(1/k), and its empirical speed depends critically on the (manually chosen) penalty value. We propose a significantly improved solver, that combines three well-established ADMM accelerators, namely, adaptive penalty, over-relaxation, and a
principled stopping criterion in a single algorithm, that is shown to significantly outperform the SotA. To prepare the problem for the use of ADMM, we introduce Q N auxiliary matrix variables Mn ∈ RIn × j̸=n Ij to represent each mode-n unfolding independently. The problem presented in (13) can therefore be rewritten as arg min λ X , {Mn }
s.t.
N X
αn ∥Mn ∥∗
(15)
where the SVT threshold τn = αn λ is fixed and independent of ρ. 3) Un -Update.: In the final step, the dual variable is updated by gradient ascent on the dual function, which yields the closed-form update
(16)
Ut+1 = Utn + X̂t(n) − Mt+1 n n .
(14)
n=1
X(n) = Mn ,
n = 1, . . . , N,
PΩ (X ) = PΩ (T ).
By the introduction of scaled dual variables Un and a penalty ρ > 0, the augmented Lagrangian is given by Lρ (X , Mn , Un ) = λ +
N X
αn ∥Mn ∥∗
(17)
n=1 N X
ρ 2 X(n) − Mn + Un F . 2 n=1
B. ADMM Subproblems
xj
N 2 ρ X xj − Mn − Un j , 2 n=1
(18)
whose exact minimizer is the consensus average N
xt+1 = j
i 1 Xh Foldn Mtn − Utn ≜ vjt , N n=1 j
(19)
where the observed entries are fixed, such that xj = [T ]j for j ∈ Ω. 2) Mn -Update.: Next, with X t+1 available the Mn subproblem is given by ρ 2 min λαn ∥Mn ∥∗ + X̂t(n) − Mn + Utn F . (20) Mn 2 Following [29], [30], we substitute the plain unfolding t+1 X(n) with the over-relaxed version t+1 X̂t(n) = ξ X(n) + (1 − ξ) Mtn
(21)
in both the Mn - and Un -updates, where ξ ∈ (1, 2) is the overrelaxation factor. Setting ξ = 1 recovers standard ADMM. For ξ ∈ [1.5, 1.8], convergence is provably maintained [29] while iteration counts are reduced by approximately 30–40% empirically. We use ξ = 1.7 throughout, following the standard recommendation of [30].
(23)
C. Adaptive Penalty Parameter Standard ADMM with fixed ρ converges at the O(1/k) rate [30], and the convergence constant depends on how well ρ balances primal and dual feasibility. We adopt the adaptive rule of [30], which updates ρ based on the primal and dual residuals computed at each iteration, such that they are defined as v uN uX 2 t X t − Mt , (24) r =t (n)
ADMM minimizes (17) by alternating exact minimizations over X , each Mn , and dual ascent on each Un . Due to the nature of the problem, all three updates can be written in closed form as follows. 1) X -Update.: Minimizing Lρ over X with Mn , Un fixed and the observation constraint (16) enforced by projection, the objective separates across tensor entries. For an unobserved entry j ∈ Ω̄, the entry xj appears in N quadratic terms (one per mode unfolding), giving the aggregate problem min
Since it was shown that the proximal operator of the nuclear norm is singular value thresholding [31], the exact closed-form solution is (22) Mt+1 = SVTτn X̂t(n) + Utn , n
n F
n=1
v uN uX 2 t tt s =ρ Mt − Mt−1 , n n
F
(25)
n=1
where the primal residual rt measures the degree of constraint violation, i.e., how far X and Mn are from consensus, while the dual residual st measures the rate at which the Mn variables are changing, i.e., the gradient of the dual function, scaled by ρ. The corresponding update rule for ρ is designed to keep rt and st within a factor of µ of each other, by increasing ρ when the primal residual is too large, and decreasing it when the dual residual is too large, such that the updated ρt+1 is given by t t t min(τ ρ , ρmax ) if r > µ s , (26) ρt+1 = max(ρt /τ, ρmin ) if st > µ rt , t ρ otherwise, with µ = 10, τ = 2, ρmin = 0.01, and ρmax = 1000 in our experiments. After each change, the dual variables are rescaled as Un ← (ρt /ρt+1 ) Un to preserve the value of the scaled augmented Lagrangian (17) across the parameter change [30]. a) Data-driven initialization.: The initial penalty ρ0 is set to balance the scale of the SVT threshold against the singular value spectrum of the initial estimate. Specifically, 0 let σ̄ denote the mean of the non-zero singular values of X(n) averaged across modes. Thus, the initial penalty is defined as σ̄ ρ0 = , (27) N ·λ so that at iteration 1 the SVT threshold τn = αn λ operates near the centre of the singular value spectrum, making the first Mn -update maximally informative. The complete procedure is summarized in Algorithm 1.
Algorithm 1 ADMM-LRTC: Low-Rank Tensor Completion via Adaptive Over-Relaxed ADMM Input: Partially observed tensor T , index set Ω, constants λ, αn , ξ, Tmax
0.11
NMSE Comparison Over Ratio of Observed Entries FaLRTC S-LRTC ADMM-LRTC
0.1 0.09
0
0.08
for n = 1, . . . , N
3: ρ0 ← σ̄ ᾱ/λ via (??) 4: for until convergence, or t = 1, 2, . . . , Tmax do 5: Compute consensus vjt via (19)
Xjt+1 ← vjt ∀j ∈ Ω̄ 7: Xjt+1 ← [T ]j ∀j ∈ Ω 8: for n = 1, . . . , N do t+1 9: X̂t(n) ← ξ X(n) + (1 − ξ) Mtn 10: Mt+1 ← SVTαn λ (X̂t(n) + Utn ) n 11: Ut+1 ← Utn + X̂t(n) − Mt+1 n n 12: end for 13: Compute rt , st via (24), (25) 14: Update ρt+1 via (26); rescale Ut+1 n 15: end for 16: return X ⋆ = clip[0,255] (X t+1 )
NMSE
1: X ← PΩ (T ) + ō PΩ̄ (1) 0 2: M0n ← SVTαn λ (X(n) ), U0n ← 0
6:
0.07 0.06 0.05 0.04 0.03 0.02 0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.6
Fig. 1. NMSE comparison of the SotA and the proposed method, with a varying ratio of observed entries in O. NMSE vs. Number of Algorithmic Iteration
0.2
FaLRTC S-LRTC ADMM-LRTC ADMM + [13]
0.18 0.16 0.14 0.12
NMSE
IV. N UMERICAL R ESULTS In this final section we compare the proposed method to the NN minimization-based SotA technique (FaLRTC) described in [13] and a more recent technique that uses a mixture of NN minimization and matrix factorization (S-LRTC), as described in [15]. For comparison, an arbitrary image1 presented in Figure 3 is chosen, which is of size 256 × 256 and contains RGB color channels represented as a third dimension, leading to a tensor of size 256 × 256 × 3. The chosen performance metric for comparison, conventionally used in the SotA, is the normalized mean square error (NMSE), defined as ∥Xb − Xtrue ∥2F . (28) NMSE = ∥Xtrue ∥2F To compare the SotA methods with the proposed method under the same conditions, identical simulation setups are chosen, where the ratio of observed values in the tensor O varies from 20% to 60%. The NMSE results over varying observation ratios are shown in Figure 1, while the convergence behavior for a fixed observation ratio of 60% is shown in Figure 2. In Figure 1, it can be observed that the proposed method clearly outperforms the SotA methods across all observation ratios. To elaborate, to reach an NMSE of 0.05, the proposed method requires only approximately 39% of the observed entries, while the SotA methods require at least 46% of the observed entries to reach the same NMSE level, which demonstrates the superior performance of the proposed method in terms of reconstruction accuracy. Next in Figure 2, the convergence behavior of the proposed method is compared to the SotA methods for a fixed observation ratio of 60%. It can be observed that the proposed method converges significantly slower than the SotA methods, however, it achieves a much lower NMSE level at convergence.
0.55
Ratio of Observed Entries
0.1 0.08 0.06 0.04 0.02 0 20
40
60
80
100
120
140
# of Iterations
Fig. 2. NMSE convergence comparison of the SotA and the proposed method, with a 60% observation ratio of O.
To further investigate the convergence behavior of the proposed method and improve its convergence speed, an additional simulation is conducted, where the proposed method is initialized with the FaLRTC SotA method reconstructed tensor, instead of a random initialization. As also illustrated in Figure 2, the proposed method with the FaLRTC initialization converges significantly faster than the proposed method with random initialization, while achieving the same NMSE level at convergence, which demonstrates that the convergence speed of the proposed method can be improved by using a better initialization. Finally, for the sake of completeness, Figure 3 shows the visual result of the SotA methods compared to the proposed method for a fixed observation ratio of 20%, along with the 1 The chosen test image is available at: https://github.com/shangqigao/TensorCompletion
Original
Sampled (20%)
FaLRTC
TMac
S-LRTC
ADMM-LRTC
Fig. 3. Illustration of the completed test image for the proposed method, compared to the original image, the sampled image and the SotA methods reconstructed images for an observation ratio of 20%.
original image and the sampled image that is used as the input for algorithms. Along the proposed method and the SotA methods shown before, the TMac method is also included in the visual comparison [14], which is a matrix factorizationbased method, but was neglected in the NMSE comparison due to its significantly worse performance compared to the other methods. V. C ONCLUSION We proposed a novel tensor completion (TC) approach in which alternating direction method of multipliers (ADMM) in addition to an adaptive penalty, as well as over-relaxation is used to solve the NN minimization problem. The resulting ADMM method is shown to significantly outperform the SotA in terms of NMSE performance, across all observation ratios, albeit at the cost of a slower convergence speed. Additionally, it is shown that the convergence speed of the proposed method can be improved by using a better initialization, such as the reconstructed tensor from the SotA method, without affecting the NMSE performance at convergence. Future work will focus on the improvement the proposed method, both in terms of NMSE performance and convergence. R EFERENCES [1] L. T. Nguyen, J. Kim, S. Kim, and B. Shim, “Localization of iot networks via low-rank matrix completion,” IEEE Transactions on Communications, vol. 67, no. 8, pp. 5833–5847, 2019. [2] Z. Chen and S. Wang, “A review on matrix completion for recommender systems,” Knowledge and Information Systems, vol. 64, no. 1, pp. 1–34, 2022. [3] E. Vlachos, G. C. Alexandropoulos, and J. Thompson, “Massive MIMO channel estimation for millimeter wave systems via matrix completion,” IEEE Signal Processing Letters, vol. 25, no. 11, pp. 1675–1679, 2018. [4] E. J. Candès and Y. Plan, “Matrix completion with noise,” Proceedings of the IEEE, vol. 98, no. 6, pp. 925–936, 2010. [5] E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Foundations of Computational Mathematics, vol. 9, no. 6, pp. 717–772, 2009. [6] E. J. Candès and T. Tao, “The power of convex relaxation: Near-optimal matrix completion,” IEEE Transactions on Information Theory, vol. 56, no. 5, pp. 2053–2080, 2010. [7] L. T. Nguyen, J. Kim, and B. Shim, “Low-rank matrix completion: A contemporary survey,” IEEE Access, vol. 7, pp. 94 215–94 237, 2019.
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