用分数正则化提升张量补全精度，更准确逼近低秩结构。
Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm
- 提出新型非凸代理函数TNPK，更好近似张量管秩
- 在真实与合成数据上均超越当前最优方法
- 算法收敛有保证，适合高维张量补全任务
本文针对低秩张量补全（LRTC）问题，提出一种新颖的非凸代理函数——张量核范数与张量Ky Fan p-k 范数之比（TNPK），以更精确地逼近张量管秩。该函数具备尺度不变性、参数灵活性及特定参数下闭式解等优点；当参数取特定值时，可退化为张量核范数与Ky Fan k范数之比（TNK）或与Frobenius范数之比（TNF）。构建了相应的LRTC模型，并在张量零空间性质（NSP）下证明低秩张量是该模型的局部极小值点。进一步推导出Ky Fan p-k逆范数的近端算子，设计了一种高效交替方向乘子法（ADMM）算法，在温和条件下保证子序列收敛。大量实验表明，该方法在合成与真实数据集上均显著优于现有先进方法。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。