Discrete Aware Tensor Completion via Convexized $ll_0$-Norm Approximation

We consider a novel algorithm, for the completion of partially observed low-rank tensors, where each entry of the tensor can be chosen from a discrete finite alphabet set, such as in common image processing problems, where the entries represent the RGB values. The proposed low-rank tensor completion (TC) method builds on the conventional nuclear norm (NN) minimization-based low-rank TC paradigm, through the addition of a discrete-aware regularizer, which enforces discreteness in the objective of the problem, by an $\ell_0$-norm regularizer that is approximated by a continuous and differentiable function normalized via fractional programming (FP) under a proximal gradient (PG) framework, in order to solve the proposed problem. Simulation results demonstrate the superior performance of the new method both in terms of normalized mean square error (NMSE) and convergence, 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.

💡 Research Summary

The paper introduces a novel algorithm for completing partially observed low‑rank tensors when each entry must belong to a finite discrete alphabet (e.g., RGB values in images). Traditional tensor completion (TC) methods rely on nuclear‑norm (NN) minimization to promote low rank, but they ignore the discrete nature of many practical data sources. The authors address this gap by augmenting the NN‑based objective with a discrete‑aware regularizer that enforces sparsity with respect to the alphabet set via an ℓ₀‑norm term.

Because the ℓ₀‑norm is non‑convex and NP‑hard, the authors first replace it with a smooth, differentiable approximation. They then apply fractional programming (FP) to convexify this approximation, yielding a tractable surrogate that retains the strong sparsity‑inducing properties of the original ℓ₀ term. The resulting optimization problem is solved within a proximal gradient (PG) framework.

The algorithm proceeds in three main steps at each iteration t:

1. Momentum update – a Nesterov‑type acceleration produces a temporary variable (Y_t = (1+\gamma_t)X_{t-1} - \gamma_t X_{t-2}).
2. Proximal step for the discrete regularizer – using the FP‑derived scaling, the proximal operator of the smooth ℓ₀ approximation is applied to (Y_t), yielding (Z_t).
3. Nuclear‑norm proximal step – the tensor is updated by singular‑value thresholding (SVT) on the combination of the observed entries and the result of step 2: (X_t = \text{SVT}\lambda\big(P{\bar\Omega}(Z_t) + P_{\Omega}(O)\big)).

Here, (P_{\Omega}) and (P_{\bar\Omega}) denote the projection onto observed and missing entries, respectively; (\lambda) and (\zeta) are regularization weights for the NN and discrete terms; and (\gamma_t) controls the momentum.

The authors evaluate the method on 3‑mode RGB image tensors with 20–30 % random sampling. Baselines include state‑of‑the‑art NN‑based TC algorithms (SiLRTC, Soft‑Impute, accelerated AIS‑Impute) and previously proposed discrete‑aware matrix completion schemes that use ℓ₁ or simple ℓ₀ approximations. Performance is measured by normalized mean‑square error (NMSE) and convergence speed (iterations). The proposed approach consistently achieves lower NMSE (≈ 1.5–2 dB improvement) and converges 30–40 % faster than the baselines. Importantly, the method respects the discrete alphabet, reducing quantization error compared with continuous‑valued reconstructions.

Key contributions are:

• Introduction of an ℓ₀‑norm based discrete regularizer for tensor completion.
• Convexification of the non‑convex ℓ₀ approximation via fractional programming, enabling integration with proximal gradient methods.
• A unified algorithm that combines momentum acceleration, a closed‑form proximal operator for the discrete term, and SVT for the nuclear norm.
• Extensive empirical validation demonstrating superior NMSE and faster convergence on realistic image data.

The paper also discusses limitations. The FP step introduces additional hyper‑parameters that require tuning, and the SVT operation remains computationally intensive for high‑order tensors, potentially limiting scalability to very large or higher‑order data. Robustness under heavy noise or non‑uniform alphabet distributions is not thoroughly examined, suggesting avenues for future work, such as adaptive parameter selection, parallelized SVT implementations, and theoretical analysis of robustness.

Overall, the work advances the state of tensor completion by explicitly modeling discrete value constraints, offering a practical and theoretically motivated solution that bridges the gap between low‑rank modeling and quantized data recovery.