通过流形融合提升高秩矩阵补全的低采样性能
High Rank Matrix Completion via Grassmannian Proxy Fusion
- 用子空间代理聚类不完整向量，基于流形距离优化
- 低采样率下性能显著优于现有方法，逼近理论采样极限
- 适合需要低样本高精度补全的科研与工程场景
本文针对高秩矩阵补全问题，提出一种基于流形代理融合的新方法。数据矩阵的列位于多个子空间的并集附近，需对列进行聚类并识别潜在子空间。现有方法普遍存在理论支持不足、结果难解释、所需样本数超过理论下限等问题。本方法通过分组子空间代理，同时最小化两个在格拉斯曼流形上的准则：(a) 每个点与其对应子空间间的弦距；(b) 所有数据点子空间间的测地距。在合成与真实数据集上的实验表明，该方法在高采样率下表现接近领先方法，在低采样率下显著更优，从而缩小了实际性能与高秩矩阵补全理论采样极限之间的差距。
原文摘要 · Abstract (English)
This paper approaches high-rank matrix completion (HRMC) by filling missing entries in a data matrix where columns lie near a union of subspaces, clustering these columns, and identifying the underlying subspaces. Current methods often lack theoretical support, produce uninterpretable results, and require more samples than theoretically necessary. We propose clustering incomplete vectors by grouping proxy subspaces and minimizing two criteria over the Grassmannian: (a) the chordal distance between each point and its corresponding subspace and (b) the geodesic distances between subspaces of all data points. Experiments on synthetic and real datasets demonstrate that our method performs comparably to leading methods in high sampling rates and significantly better in low sampling rates, thus narrowing the gap to the theoretical sampling limit of HRMC.
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