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.

中文翻译：

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稀疏子空间聚类的高效求解器

稀疏子空间聚类 (SSC) 是机器学习和计算机视觉中的一种流行方法，用于聚类位于低维线性或仿射子空间联合附近的 $n$ 个数据点。Elhamifar 和 Vidal 引入的标准模型将每个数据点表示为其他数据点的稀疏线性或仿射组合，使用 $\ell_1$ 或 $\ell_0$ 正则化来加强稀疏性。$\ell_1$ 模型是凸的，具有理论上的保证，但需要 $O(n^2)$ 存储，通常通过乘法器的交替方向法 (ADMM) 求解，该方法需要 $O(n^3)$ 次触发器. $\ell_0$ 模型是大 $n$ 的首选模型，因为它只需要 $n$ 中的线性内存，通常通过正交匹配追踪 (OMP) 解决，无法处理仿射子空间的情况。我们的第一个贡献是展示如何使用矩阵求逆引理修改 ADMM 以采用 $O(n^2)$ 触发器而不是 $O(n^3)$。然后，我们的主要贡献是表明近端梯度框架可以解决 SSC，涵盖 $\ell_1$ 和 $\ell_0$ 模型，以及线性和仿射约束。对于 $\ell_1$ 和 $\ell_0$，具有仿射约束的邻近算子是非平凡的，因此我们推导出有效的邻近算子。在 $\ell_1$ 的情况下，我们的方法只需要 $O(n^2)$ 次触发器，而在 $\ell_0$ 的情况下，内存在 $n$ 中是线性的。这是第一个结合仿射约束解决 $\ell_0$ 问题的算法。合成数据和真实数据的数值实验表明，基于近端梯度的求解器是最先进的，但更重要的是，