在任意强但局部相关的噪声下恢复正交张量
Recovering orthogonal tensors under arbitrarily strong, but locally correlated, noise
摘要：
我们考虑的问题，恢复正交可分张量与任意大幅度的噪声失真的元素的子集。我们专注于分解中的每个模式被具有局部相关的分量的噪声向量破坏的特定情况，即，附近的组件。我们表明，这个确定性张量完成问题具有不寻常的属性，它可以在多项式时间内解决，如果张量的秩是足够大的。这与典型的低秩张量和矩阵完备设置的低秩假设截然相反。我们表明，我们的问题可以通过一个系统的耦合西尔维斯特方程，并显示如何加速其解决方案的交替求解。这使得即使具有大量缺失条目也能够恢复，例如对于具有多达$40\%$缺失条目的秩$n$的$n$维张量。
英文摘要：
We consider the problem of recovering an orthogonally decomposable tensor with a subset of elements distorted by noise with arbitrarily large magnitude. We focus on the particular case where each mode in the decomposition is corrupted by noise vectors with components that are correlated locally, i.e., with nearby components. We show that this deterministic tensor completion problem has the unusual property that it can be solved in polynomial time if the rank of the tensor is sufficiently large. This is the polar opposite of the low-rank assumptions of typical low-rank tensor and matrix completion settings. We show that our problem can be solved through a system of coupled Sylvester-like equations and show how to accelerate their solution by an alternating solver. This enables recovery even with a substantial number of missing entries, for instance for $n$-dimensional tensors of rank $n$ with up to $40\%$ missing entries.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
