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.

中文翻译：

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颤抖表示的同时鲁棒子空间恢复和半稳定性

我们考虑在给定的 $m$-元组中同时找到低维子空间结构的问题，该元组可能损坏的高维数据集都相同大小。我们将这个问题称为同时鲁棒子空间恢复（SRSR），并提供了一种颤动不变的理论方法。我们展示了 SRSR 是更普遍问题的一个特例，即有效地确定颤动表示是否是半稳定的（在几何不变理论的意义上），如果不是，则找到以最佳方式证明的子表示表示不是半稳定的。在本文中，我们表明可以有效地解决 SRSR 和更一般的颤动半稳定问题。