张量正交子空间拆分：理论与应用

中文摘要 AI 辅助

张量表示已成为建模多维数据的基础范式，可保留多模式间的内在关联。本文提出一种名为张量正交子空间拆分（Tensor Orthogonal Subspace Split, TOSS）的新理论框架，该框架沿指定模式将张量显式拆分为两个正交分量：位于指定子空间的主导分量，以及位于对应正交补空间的残差分量。我们首先给出TOSS的一般形式，并系统研究其基本性质。作为重要且具有实际意义的特例，我们进一步引入秩1 TOSS，其在拆分模式上施加可分离的秩1结构，且具有清晰的几何解释。该形式可自然捕捉主导的一致模式，同时有效分离正交残差分量。所提框架为张量域正交拆分建立了统一的理论基础，为各类应用中的结构化张量建模开辟了新途径。基于已发展的TOSS理论，我们选取高光谱图像复原与彩色视频背景建模作为两个代表性任务，针对这些任务构建了对应的优化模型，并开发了高效算法以求解所得问题。大量实验结果验证了所提方法的有效性与优越性。

英文摘要

Tensor representations have emerged as a fundamental paradigm for modeling multidimensional data by preserving intrinsic correlations across multiple modes. This paper proposes a novel theoretical framework, termed Tensor Orthogonal Subspace Split (TOSS), which explicitly splits a tensor, along a prescribed mode, into two orthogonal components: a dominant component lying in a prescribed subspace and a residual component lying in the corresponding orthogonal complement. We first present the general formulation of TOSS and systematically investigate its fundamental properties. As an important and practically meaningful special case, we further introduce the rank-one TOSS, which imposes a separable rank-one structure along the splitting mode and admits a clear geometric interpretation. This formulation naturally captures dominant consistent patterns while effectively isolating orthogonal residual component. The proposed framework establishes a unified theoretical foundation for tensor-domain orthogonal split and opens new avenues for structured tensor modeling across diverse applications. Building upon the developed TOSS theory, hyperspectral image restoration and color video background modeling are considered as two representative tasks, for which corresponding optimization models are formulated. Efficient algorithms are developed to solve the resulting problems. Extensive experimental results validate the effectiveness and superiority of the proposed approaches.

发表机构

• Yunnan University（云南大学）
• Anhui Jianzhu University（安徽建筑大学）
• Hong Kong Baptist University（香港浸会大学）
• University of Macau（澳门大学）

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