几何依赖的匹配追踪：线性回归与 LASSO 收敛性的过渡阶段
Geometry-dependent matching pursuit: a transition phase for convergence on linear regression and LASSO
AI总结：
本文提出依据问题几何结构生成正则化匹配追踪算法的原则性方法，给出收敛保证，并揭示其从欠参数化到过参数化情形的收敛过渡现象。
摘要：
贪婪一阶方法，例如采用 Gauss-Southwell 规则的坐标下降法或匹配追踪，因其天然倾向于给出稀疏解以及精细的收敛保证，已在优化中受到欢迎。在本工作中，我们提出一种有原则的方法，用于生成适配当前问题几何结构的（正则化）匹配追踪算法，并给出其收敛保证。基于这些结果，我们推导近似收敛保证，并描述（正则化）匹配追踪在欠参数化模型到过参数化模型之间收敛时的一种过渡现象。
英文摘要：
Greedy first-order methods, such as coordinate descent with Gauss-Southwell rule or matching pursuit, have become popular in optimization due to their natural tendency to propose sparse solutions and their refined convergence guarantees. In this work, we propose a principled approach to generating (regularized) matching pursuit algorithms adapted to the geometry of the problem at hand, as well as their convergence guarantees. Building on these results, we derive approximate convergence guarantees and describe a transition phenomenon in the convergence of (regularized) matching pursuit from underparametrized to overparametrized models.
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