论文标题
Riemannian优化方法聚类问题
A Riemannian Optimization Approach to Clustering Problems
论文作者
论文摘要
本文以$ \ min_ {x \ in \ mathcal {f} _v} f（x） +λ\ | x \ | _1，$ f $平滑，$ \ nathcal {f} _v = \ in \ in \ in \ in \ ntipe} v \ in \ mathrm {span}（x）\} $，$ v $是给定的正值向量。聚集模型，包括但不限于$ k $ -MEANS，社区检测和归一化切割使用的模型，可以作为优化问题进行重新校正。事实证明，域$ \ MATHCAL {f} _v $形成一个紧凑的嵌入式submanifold的$ \ mathbb {r}^{r}^{n \ times q} $和优化相关的工具，包括计算上有效的缩回和$ \ nater n Mathcal的正常空间的计算家族。提出了一种允许自适应步进大小的不精确的加速riemannian近端方法，并建立了其全局收敛。用于网络中的社区检测和图像分割的归一化切割的数值实验用于证明该方法的性能。
This paper considers the optimization problem in the form of $\min_{X \in \mathcal{F}_v} f(x) + λ\|X\|_1,$ where $f$ is smooth, $\mathcal{F}_v = \{X \in \mathbb{R}^{n \times q} : X^T X = I_q, v \in \mathrm{span}(X)\}$, and $v$ is a given positive vector. The clustering models including but not limited to the models used by $k$-means, community detection, and normalized cut can be reformulated as such optimization problems. It is proven that the domain $\mathcal{F}_v$ forms a compact embedded submanifold of $\mathbb{R}^{n \times q}$ and optimization-related tools including a family of computationally efficient retractions and an orthonormal basis of any normal space of $\mathcal{F}_v$ are derived. An inexact accelerated Riemannian proximal gradient method that allows adaptive step size is proposed and its global convergence is established. Numerical experiments on community detection in networks and normalized cut for image segmentation are used to demonstrate the performance of the proposed method.
