Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm

2026-06-17

AI Summary

This work addresses the low-rank tensor completion problem by proposing a novel non-convex regularizer—the ratio of the tensor nuclear norm to the Ky Fan $p$-$k$ norm (TNPK)—to more accurately approximate the tensor tubal rank. The proposed regularizer enjoys scale invariance, parameter flexibility, and admits a closed-form solution under certain conditions, with the tensor nuclear norm (TNK) and tensor Ky Fan norm (TNF) as its special cases. Leveraging properties of the tensor null space, the authors theoretically establish that low-rank tensors correspond to local minima of the proposed model. For optimization, they design a proximal operator based on the inverse Ky Fan $p$-$k$ norm and integrate it into an ADMM algorithm with guaranteed subsequential convergence. Extensive experiments demonstrate that the method significantly outperforms state-of-the-art approaches on both synthetic and real-world datasets.

Abstract

This paper addresses low-rank tensor completion (LRTC) by proposing a novel nonconvex surrogate, namely the ratio of the tensor nuclear norm to the tensor Ky Fan p-k norm (TNPK), to accurately approximate the tensor tubal rank. The TNPK possesses appealing properties, including scale invariance, parameter flexibility, and the existence of closed-form solutions under specific choices of p and k. With specific parameter settings of p and k, it reduces to the ratio of the tensor nuclear norm to the tensor Ky Fan k norm (TNK) or the ratio of the tensor nuclear norm to the tensor Frobenius norm (TNF). We construct a LRTC model and, under the tensor null space property (NSP), prove that low-rank tensors are local minimizers of the proposed model. Moreover, we derive the proximal operator of the Ky Fan p-k inverse-norm and further develop an efficient alternating direction method of multipliers (ADMM) algorithm with guaranteed subsequential convergence under mild conditions. Extensive experiments on synthetic and real-world datasets validate the superior performance of our method against state-of-the-art competitors.

Problem

• Low-Rank Tensor Completion
• Tensor Tubal Rank
• Ky Fan p-k Norm
• Nonconvex Surrogate

Innovation

• low-rank tensor completion
• fractional regularization
• Ky Fan p-k norm
• nonconvex surrogate
• ADMM algorithm