We consider the problem of simultaneously finding lower-dimensional subspace structures in a given $m$-tuple of possibly corrupted, high-dimensional data sets all of the same size. We refer to this problem as simultaneous robust subspace recovery (SRSR) and provide a quiver invariant theoretic approach to it. We show that SRSR is a particular case of the more general problem of effectively deciding whether a quiver representation is semi-stable (in the sense of Geometric Invariant Theory) and, in case it is not, finding a subrepresentation certifying in an optimal way that the representation is not semi-stable. In this paper, we show that SRSR and the more general quiver semi-stability problem can be solved effectively.
翻译:我们考虑了在某个可能损坏的、高维的数据集中同时找到某个特定百万美元的低维次空间结构的问题,我们将此问题称为同时稳健的子空间回收(SRSR),并提供了一种变化不定的理论方法。我们表明,SR是一个比较普遍的问题,即有效确定一个松动的表示是否半可(从几何变量理论的意义上)是半可分布的,如果不是,则以最佳的方式找到一个子代表的证明,说明其表示不是半稳定的。我们在本文件中表明,SR和较普遍的松动半稳定问题可以有效解决。