论文标题
追求网络套索的群集结构：恢复条件和非凸口扩展
Pursuit of the Cluster Structure of Network Lasso: Recovery Condition and Non-convex Extension
论文作者
论文摘要
网络套索（简称NL）是一种通过同时聚集数据样本并将模型拟合到样本的方法来估算模型的方法。由于使用了正规器的凸度，因此它经常成功地形成簇，但由于其凸面的凸度，因此可能存在限制。本文着重于NL通过开发非convex扩展而产生和加强的群集结构，我们将其称为网络修剪套索（简称NTL）。具体而言，我们首先研究了足够的条件，可以确保根据Sun等人的结果恢复NL的潜在簇结构。 （2021）对于凸聚类，这是用于聚类的NL的特殊情况。其次，我们将NL扩展到NTL，以将基数（或，$ \ ell_0 $ - ）结合起来，并重写用$ \ ell_0 $ norm定义的约束优化问题（一个不连续的功能），为等效不约束的不约束的连续优化问题。我们开发了ADMM算法来求解NTL并提供其收敛结果。数值插图表明，当NL未能形成簇而不合并相关参数的先验知识时，非convex扩展提供了更清晰的群集结构。
Network Lasso (NL for short) is a methodology for estimating models by simultaneously clustering data samples and fitting the models to the samples. It often succeeds in forming clusters thanks to the geometry of the $\ell_1$-regularizer employed therein, but there might be limitations because of the convexity of the regularizer. This paper focuses on the cluster structure that NL yields and reinforces it by developing a non-convex extension, which we call Network Trimmed Lasso (NTL for short). Specifically, we first study a sufficient condition that guarantees the recovery of the latent cluster structure of NL on the basis of the result of Sun et al. (2021) for Convex Clustering, which is a special case of NL for clustering. Second, we extend NL to NTL to incorporate a cardinality (or, $\ell_0$-)constraint and rewrite the constrained optimization problem defined with the $\ell_0$ norm, a discontinuous function, into an equivalent unconstrained continuous optimization problem. We develop ADMM algorithms to solve NTL and provide its convergence results. Numerical illustrations demonstrate that the non-convex extension provides a more clear-cut cluster structure when NL fails to form clusters without incorporating prior knowledge of the associated parameters.
