标题： 一个用于回归的贪心同伦方法非凸约束 显示英文标题

标题： A Greedy Homotopy Method for Regression with Nonconvex Constraints

摘要： 约束最小二乘回归是高维数据分析的重要工具。 给定一个输入变量的划分 $\mathcal{G}$，本文考虑了一类特定的非凸约束函数，这些函数鼓励线性模型从 $\mathcal{G}$ 中的一小部分组中选择少量变量。 这类约束在许多实际应用中具有相关性，例如全基因组关联研究（GWAS）。 受Lasso同伦方法效率的启发，我们提出了RepLasso，这是一种贪婪同伦算法，通过求解一系列适配的凸替代问题来尝试解决由此产生的非凸问题序列。 我们证明了在某些情况下，RepLasso能够恢复非凸问题的全局最小值。 此外，即使不能恢复全局最小值，我们也证明在相关情况下，它在支持和符号支持恢复方面不会比Lasso差，并且在实践中优于Lasso。 我们通过实证表明，该策略也可以用于改进其他类似Lasso的算法。 最后，强直性脊柱炎的GWAS突显了我们方法的实际效用。 显示英文摘要

摘要： Constrained least squares regression is an essential tool for high-dimensional data analysis. Given a partition $\mathcal{G}$ of input variables, this paper considers a particular class of nonconvex constraint functions that encourage the linear model to select a small number of variables from a small number of groups in $\mathcal{G}$. Such constraints are relevant in many practical applications, such as Genome-Wide Association Studies (GWAS). Motivated by the efficiency of the Lasso homotopy method, we present RepLasso, a greedy homotopy algorithm that tries to solve the induced sequence of nonconvex problems by solving a sequence of suitably adapted convex surrogate problems. We prove that in some situations RepLasso recovers the global minima of the nonconvex problem. Moreover, even if it does not recover global minima, we prove that in relevant cases it will still do no worse than the Lasso in terms of support and signed support recovery, while in practice outperforming it. We show empirically that the strategy can also be used to improve over other Lasso-style algorithms. Finally, a GWAS of ankylosing spondylitis highlights our method's practical utility.