Variable Selection via Inverse Modeling for Detecting Nonlinear Relationships

随着许多应用领域中数据量的不断增长，对检测影响响应变量值的因素的有效方法的需求很高。它是越来越重要的发展方法来检测变量施加显着的非线性响应。受20世纪90年代初发展起来的切片逆回归方法的启发，PI提出了一个在高维非线性系统中开发有效变量选择策略的一般框架。PI将进一步研究这些变量选择算法的理论特性。建议的理论研究将提供现有的降维技术的局限性时，随着样本量的维数增长的理论理解。 随着在许多应用领域中数据量的不断增长，用于检测可能影响感兴趣的目标量（响应变量）的值的因素的有效方法的需求很高。这个问题在回归建模和统计学习中被称为“变量（或特征）选择”，是统计和机器学习中的一个长期问题。PI在此侧重于检测可能对响应变量产生非线性和/或交互影响的因素。PI小组最近的研究表明，切片逆回归（SIR）和逆建模策略提供了一个强大的框架，在高维非线性系统中开发有效的变量选择策略。PI旨在开发更强大和有效的工具来检测这种复杂的关系，并研究基于SIR的算法的理论特性。所提出的方法也将适用于做强大的变量选择分类问题。这些理论研究将提供：（a）当维数随样本量增长时，现有降维技术的局限性的理论认识;（B）指导构建必要的稀疏性条件，以保证超高维非线性问题中变量选择的一致性;（c）在这种设置下，最佳学习算法可以达到的最佳收敛速度;以及（d）所提出的算法是否能够实现或离最优性不远的理论证明。

With the ever-growing amount of data in many application areas, effective methods for detecting factors influencing the value of a response variable are in high demand. It is of growing importance to develop methods for detecting variables that exert significant nonlinear response. Inspired by the sliced inverse regression method developed in the early 1990s, the PI proposes a general framework for developing effective variable selection strategies in nonlinear systems of high dimension. The PI will further study theoretical properties of these variable selection algorithms. The proposed theoretical investigation will provide theoretical understanding of limitations of existing dimension-reduction techniques when the dimensionality grows with the sample size. With the ever-growing amount of data in many application areas, effective methods for detecting factors that may influence the value of a target quantity of interest (response variable) are in high demand. The problem is termed as "variable (or feature) selection" in regression modeling and statistical learning, and is a long-standing problem in statistics and machine learning. The PI focuses here on the detection of factors that may exert nonlinear and/or interactive effects on the response variable. Recent studies from the PI's group reveal that the sliced inverse regression (SIR) and inverse modeling strategies provide a powerful framework for developing effective variable selection strategies in nonlinear systems of high dimension. The PI aims at developing more robust and effective tools for detecting such complex relationships and studying theoretical properties of SIR-based algorithms. The proposed method will also be applicable to do robust variable selection for classification problems. The proposed theoretical investigations will provide (a) theoretical understanding of limitations of existing dimension-reduction techniques when the dimensionality grows with the sample size; (b) guidance on the construction of necessary sparsity conditions that can guarantee consistency of variable selections in ultra-high dimensional nonlinear problems; (c) the optimal convergence rate of that the best possible learning algorithm can achieve in such settings; and (d) theoretical justifications whether the proposed algorithms can achieve or are not far from the optimality.