Subspace clustering is the unsupervised grouping of points lying near a union of low-dimensional linear subspaces. Algorithms based directly on geometric properties of such data tend to either provide poor empirical performance, lack theoretical guarantees, or depend heavily on their initialization. We present a novel geometric approach to the subspace clustering problem that leverages ensembles of the K-subspaces (KSS) algorithm via the evidence accumulation clustering framework. Our algorithm, referred to as ensemble K-subspaces (EKSS), forms a co-association matrix whose (i,j)th entry is the number of times points i and j are clustered together by several runs of KSS with random initializations. We prove general recovery guarantees for any algorithm that forms an affinity matrix with entries close to a monotonic transformation of pairwise absolute inner products. We then show that a specific instance of EKSS results in an affinity matrix with entries of this form, and hence our proposed algorithm can provably recover subspaces under similar conditions to state-of-the-art algorithms. The finding is, to the best of our knowledge, the first recovery guarantee for evidence accumulation clustering and for KSS variants. We show on synthetic data that our method performs well in the traditionally challenging settings of subspaces with large intersection, subspaces with small principal angles, and noisy data. Finally, we evaluate our algorithm on six common benchmark datasets and show that unlike existing methods, EKSS achieves excellent empirical performance when there are both a small and large number of points per subspace.

翻译：子空间群集是位于低维线性子空间联盟附近的不受监督的一组点。 直接以这些数据的几何属性为根据的数值, 往往提供不甚实的经验性、 缺乏理论保证或严重依赖初始化。 我们对子空间群集问题提出了一种新的几何方法, 通过证据积累组合框架, 利用K子空间( KSS) 算法的组合, 利用对齐绝对内产物的单向转换。 我们的算法称为共振 K子空间( KSS), 形成一个共振组合矩阵, 其( i, j) 分数的输入点是i和 j 由几批带有随机初始化的 KSS 组合成。 我们证明, 任何构成接近于对齐绝对内产物进行单向转换的亲近性矩阵的算法, 其具体例子显示, EKSS 与该表的相近性矩阵, 因此我们提议的算法可以在类似的条件下回收子空间空间空间段的次空间段数, 其(i, j) 分数的分数, 分数分数分数分数的分数,, 分数分数分数的分数分数分数分数的分数分数,, 分数分数分数分数分数的分数分数分数分数分数分数分数分数是由由由几时间空间空间空间点的数值由数由由由由由由几行的数值由数由数由几运行的数值组合的计算法,,, 混合法, 我们算算算法,, 我们的算法,, 和最后的算法, 我们的算法, 和最后的算法, 我们的算法, 我们的现有数据的算法, 我们现有的数据的多数级法,, 和最后的进度性数据组法,,,,, 和最后的顺序是我们现有的数据组法,, 我们的恢复算法, 我们的顺序的顺序的顺序的顺序的 和交错算方法, 和交法,, 我们的恢复法, 我们的恢复法, 最优, 和交错法, 和交错算, 最优的比,,,, 我们的