A Nullspace Property for Subspace-Preserving Recovery
A Nullspace Property for Subspace-Preserving Recovery
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发表时间：
2021
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通讯作者：
M. Kaba;Chong You;Daniel P. Robinson;Enrique Mallada;R. Vidal
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作者：
M. Kaba;Chong You;Daniel P. Robinson;Enrique Mallada;R. Vidal
经典稀疏恢复的大部分理论都是基于字典上既必要又足够的条件（例如，零空间属性）或仅足够（例如，不相干和限制等距）。相比之下，子空间保持恢复的大部分理论，稀疏子空间分类和聚类方法的理论基础，是基于子空间和数据上的条件，这些条件仅是足够的（例如，子空间不相干性和数据内径）。本文从经典的零空间性质出发，导出了子空间保持恢复的一个充要条件.基于这个新的条件，这里称为子空间零空间属性，我们得到等价的特征，要么承认一个明确的几何解释，数据分布和子空间分离的恢复成功，或可以验证艾德使用一个适当定义的集的极值点的有限集。我们进一步利用这些特征来导出新的充分条件，基于内径和外径测度和对偶边界，推广现有条件并保留几何解释。这些结果填补了子空间保持恢复文献中的一个重要空白。
Much of the theory for classical sparse recovery is based on conditions on the dictionary that are both necessary and sufficient (e.g., nullspace property) or only sufficient (e.g., incoherence and restricted isometry). In contrast, much of the theory for subspace-preserving recovery , the theoretical underpinnings for sparse subspace classification and clustering methods, is based on conditions on the subspaces and the data that are only suffi-cient (e.g., subspace incoherence and data inner-radius). This paper derives a necessary and suffi-cient condition for subspace-preserving recovery that is inspired by the classical nullspace property. Based on this novel condition, called here the sub-space nullspace property , we derive equivalent characterizations that either admit a clear geometric interpretation that relates data distribution and subspace separation to the recovery success, or can be verified using a finite set of extreme points of a properly defined set. We further exploit these characterizations to derive new sufficient conditions, based on inner-radius and outer-radius measures and dual bounds, that generalize existing conditions and preserve the geometric interpretations. These results fill an important gap in the subspace-preserving recovery literature.