利用子空间信息的张量补全

中文摘要 AI 辅助

张量补全在应用和理论研究中都引起了广泛关注。在标准均匀采样下，现有的多项式时间保证通常需要比自由度数量更多的观测，这促使人们研究在高缺失率场景中可能存在的统计-计算差距。幸运的是，在许多实际场景中，可以获得辅助信息，这些信息可以提供有价值的见解以缓解这些挑战。在本文中，我们介绍了一种名为利用子空间信息的张量补全（TCSI）的算法，该算法通过估计的子空间来整合辅助信息。我们的方法首先从可用的辅助信息中提取子空间，然后将张量补全重新表述为一个矩阵回归问题。我们提供的理论分析表明，当存在准确的子空间信息时，所需的样本复杂度降低到未耦合环境维度中接近线性的阶数，从而将耦合模式维度从主导项中移除。利用估计的子空间信息，我们获得了比现有几种被动均匀采样保证更宽松的充分信噪比要求。在额外的温和条件下，我们获得了更精确的统计误差界。我们的理论发现得到了数值模拟的支持。我们将TCSI应用于全球总电子含量（TEC）地图的重建，并在实验中观察到比所比较方法更低的重建误差。

英文摘要

Tensor completion has attracted significant attention in both applications and theoretical research. Under standard uniform sampling, existing polynomial-time guarantees generally require more observations than the number of degree of freedom, motivating the study of a possible statistical-to-computational gap in highly missing regimes. Fortunately, in many practical scenarios, side information is available, which can provide valuable insights to mitigate these challenges. In this paper, we introduce an algorithm called Tensor Completion using Subspace Information (TCSI) that incorporates side information through an estimated subspace. Our approach first extracts the subspace from the available side information and then reformulates tensor completion as a matrix regression problem. We provide a theoretical analysis showing that, when accurate subspace information is available, the required sample complexity is reduced to nearly linear order in the uncoupled ambient dimensions, removing the coupled-mode dimension from the leading term. Leveraging the estimated subspace information, we obtain a less stringent sufficient signal-to-noise ratio requirement than those in several existing passive-uniform-sampling guarantees. Under additional mild conditions, we obtain a sharper statistical error bound. Our theoretical findings are supported by numerical simulations. We apply TCSI to the reconstruction of global Total Electron Content (TEC) maps and observe lower reconstruction errors than the compared methods in our experiments.

发表机构

• Fudan University（复旦大学）
• Hong Kong Baptist University（香港浸会大学）

机构由 AI 辅助整理，请以论文原文为准。