A number of approaches to graph-based subspace clustering, which assumes that the clustered data points were drawn from an unknown union of multiple subspaces, have been proposed in recent years. Despite their successes in computer vision and data mining, most neglect to simultaneously consider global and local information, which may improve clustering performance. On the other hand, the number of connected components reflected by the learned affinity matrix is commonly inconsistent with the true number of clusters. To this end, we propose an adaptive affinity matrix learning method, nonnegative self-representation with a fixed rank constraint (NSFRC), in which the nonnegative self-representation and an adaptive distance regularization jointly uncover the intrinsic structure of data. In particular, a fixed rank constraint as a prior is imposed on the Laplacian matrix associated with the data representation coefficients to urge the true number of clusters to exactly equal the number of connected components in the learned affinity matrix. Also, we derive an efficient iterative algorithm based on an augmented Lagrangian multiplier to optimize NSFRC. Extensive experiments conducted on real-world benchmark datasets demonstrate the superior performance of the proposed method over some state-of-the-art approaches.

近年来，已经提出了许多基于图的子空间聚类的方法，这些方法假定聚类的数据点是从多个子空间的未知联合中得出的。尽管他们在计算机视觉和数据挖掘方面取得了成功，但大多数人忽略了同时考虑全局和本地信息，这可能会改善群集性能。另一方面，学习的亲和力矩阵反映的连接组件的数量通常与群集的真实数量不一致。为此，我们提出了一种自适应亲和矩阵学习方法，即具有固定秩约束的非负自表示（NSFRC），其中非负自表示和自适应距离正则化共同揭示了数据的内在结构。特别是，在与数据表示系数相关联的拉普拉斯矩阵上施加一个先验的固定秩约束，以促使簇的真实数量精确地等于学习的亲和力矩阵中连接组件的数量。此外，我们基于增强的拉格朗日乘数推导了一种有效的迭代算法，以优化NSFRC。在现实世界的基准数据集上进行的大量实验证明，该方法优于某些最新方法。