This paper studies the subspace clustering problem. Given some data points approximately drawn from a union of subspaces, the goal is to group these data points into their underlying subspaces. Many subspace clustering methods have been proposed and among which sparse subspace clustering and low-rank representation are two representative ones. Despite the different motivations, we observe that many existing methods own the common block diagonal property, which possibly leads to correct clustering, yet with their proofs given case by case. In this work, we consider a general formulation and provide a unified theoretical guarantee of the block diagonal property. The block diagonal property of many existing methods falls into our special case. Second, we observe that many existing methods approximate the block diagonal representation matrix by using different structure priors, e.g., sparsity and low-rankness, which are indirect. We propose the first block diagonal matrix induced regularizer for directly pursuing the block diagonal matrix. With this regularizer, we solve the subspace clustering problem by Block Diagonal Representation (BDR), which uses the block diagonal structure prior. The BDR model is nonconvex and we propose an alternating minimization solver and prove its convergence. Experiments on real datasets demonstrate the effectiveness of BDR.

中文翻译：

---

通过块对角线表示进行子空间聚类


本文研究子空间聚类问题。给定一些近似从子空间并集得出的数据点，目标是将这些数据点分组到其底层子空间中。已经提出了许多子空间聚类方法，其中稀疏子空间聚类和低秩表示是两种代表性的方法。尽管动机不同，但我们观察到许多现有方法拥有共同的块对角线属性，这可能导致正确的聚类，但它们的证明是根据具体情况给出的。在这项工作中，我们考虑了一个通用的公式，并为块对角线性质提供了统一的理论保证。许多现有方法的块对角线属性属于我们的特殊情况。其次，我们观察到许多现有方法通过使用不同的结构先验（例如稀疏性和低秩性）来近似块对角表示矩阵，这些都是间接的。我们提出了第一个块对角矩阵诱导正则化器，用于直接追求块对角矩阵。有了这个正则化器，我们通过块对角表示（BDR）解决子空间聚类问题，它使用块对角线结构先验。 BDR 模型是非凸的，我们提出了一种交替最小化求解器并证明了其收敛性。在真实数据集上的实验证明了 BDR 的有效性。