This thesis investigates robust and flexible methods for matrix and tensor analysis, which are fundamental in data science. The primary focus of this work is the development of Guaranteed Sampling Flexibility for Low-Tubal-Rank Tensor Completion, a project aimed at addressing the limitations of existing sampling methods for tensor completion, such as Bernoulli and t-CUR sampling, which often lack flexibility across diverse applications.To overcome these challenges, we introduce Tensor Cross-Concentrated Sampling (t-CCS), an extension of the matrix Cross-Concentrated Sampling (CCS) model to tensors, and propose a novel non-convex algorithm, Iterative Tensor CUR Completion (ITCURC), specifically tailored for t-CCS-based tensor completion. Theoretical analysis provides sufficient conditions for low-rank tensor recovery and presents a detailed sampling complexity analysis. These findings are further validated through extensive testing on both real-world and synthetic datasets. In addition to the main project, this thesis includes another one complementary study. The study explores the robustness of CCS model for matrix completion, a recent approach demonstrated to effectively capture cross-concentrated data dependencies. However, its robustness to sparse outliers has remained underexplored. To address this gap, we propose the Robust CCS Completion problem and develop a non-convex iterative algorithm, Robust CUR Completion (RCURC). Empirical results on synthetic and real-world datasets demonstrate that RCURC is both efficient and robust against outliers, making it a powerful tool for recovering incomplete data. Collectively, these projects advance the robustness and flexibility of matrix and tensor methods, enhancing their applicability in complex, real-world data environments.
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