While Bernoulli sampling is extensively studied in tensor completion, t-CUR sampling approximates low-tubal-rank tensors via lateral and horizontal subtensors. However, both methods lack sufficient flexibility for diverse practical applications. To address this, we introduce Tensor Cross-Concentrated Sampling (t-CCS), a novel and straightforward sampling model that advances the matrix cross-concentrated sampling concept within a tensor framework. t-CCS effectively bridges the gap between Bernoulli and t-CUR sampling, offering additional flexibility that can lead to computational savings in various contexts. A key aspect of our work is the comprehensive theoretical analysis provided. We establish a sufficient condition for the successful recovery of a low-rank tensor from its t-CCS samples. In support of this, we also develop a theoretical framework validating the feasibility of t-CUR via uniform random sampling and conduct a detailed theoretical sampling complexity analysis for tensor completion problems utilizing the general Bernoulli sampling model. Moreover, we introduce an efficient non-convex algorithm, the Iterative t-CUR Tensor Completion (ITCURTC) algorithm, specifically designed to tackle the t-CCS-based tensor completion. We have intensively tested and validated the effectiveness of the t-CCS model and the ITCURTC algorithm across both synthetic and real-world datasets.

在张量完成中，我们引入了一种新颖而简单的采样模型，Tensor Cross-Concentrated Sampling (t-CCS)，有效地弥补了Bernoulli和t-CUR采样之间的差距，并具有更大的灵活性。同时，我们提供了详尽的理论分析，建立了低秩张量从t-CCS采样中成功恢复的充分条件，并且设计了适用于基于t-CCS的张量完成问题的高效非凸算法，即Iterative t-CUR Tensor Completion (ITCURTC)算法。我们对t-CCS模型和ITCURTC算法在合成和真实数据集上进行了大量测试和验证。

低管狀秩張量補全的保證採樣靈活性