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.

我们提出了一种基于集合 K 子空间 (EKSS) 的几何方法，该算法通过证据积累聚类框架，形成一个共同关联矩阵，该矩阵的 (i，j) th 项是由多次运行具有随机初始化的 KSS 聚类算法将点 i 和 j 配对在一起的次数。我们表明，当一个算法形成的亲和力矩阵的具有和绝对内积的单调变换接近的条目时，该算法可以提供恢复保证。