Sparse subspace clustering (SSC) is a popular method in machine learning and computer vision for clustering $n$ data points that lie near a union of low-dimensional linear or affine subspaces. The standard models introduced by Elhamifar and Vidal express each data point as a sparse linear or affine combination of the other data points, using either $\ell_1$ or $\ell_0$ regularization to enforce sparsity. The $\ell_1$ model, which is convex and has theoretical guarantees but requires $O(n^2)$ storage, is typically solved by the alternating direction method of multipliers (ADMM) which takes $O(n^3)$ flops. The $\ell_0$ model, which is preferred for large $n$ since it only needs memory linear in $n$, is typically solved via orthogonal matching pursuit (OMP) and cannot handle the case of affine subspaces. Our first contribution is to show how ADMM can be modified using the matrix-inversion lemma to take $O(n^2)$ flops instead of $O(n^3)$. Then, our main contribution is to show that a proximal gradient framework can solve SSC, covering both $\ell_1$ and $\ell_0$ models, and both linear and affine constraints. For both $\ell_1$ and $\ell_0$, the proximity operator with affine constraints is non-trivial, so we derive efficient proximity operators. In the $\ell_1$ case, our method takes just $O(n^2)$ flops, while in the $\ell_0$ case, the memory is linear in $n$. This is the first algorithm to solve the $\ell_0$ problem in conjunction with affine constraints. Numerical experiments on synthetic and real data demonstrate that the proximal gradient based solvers are state-of-the-art, but more importantly, we argue that they are more convenient to use than ADMM-based solvers because ADMM solvers are highly sensitive to a solver parameter that may be data set dependent.

该论文提出一种基于近端梯度框架的方法，可以解决支持线性和仿射约束的稀疏子空间聚类问题，同时还提供了有效的算法以及低存储开销的解决方案。实验证明，该算法不依赖于敏感的正则化参数，对于稀疏误差和高噪声的情况下也更不敏感。

稀疏子空间聚类的高效求解器