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摘要: 维数约简作为机器学习的经典问题之一，主要用于处理维数灾问题、帮助加速算法的计算效率和提高可解释性以及数据可视化.传统的维数约简算法如主成分分析（Principal component analysis，PCA）和线性判别分析等只能处理无标签数据或者分类数据.然而，当预测变量为一元或多元连续型实值变量时，这些处理无标签数据或分类数据的维数约简方法则不能形成有效的预测性能.近20年来，有一系列工作从多个角度对这一问题展开了研究，并取得了系统性的研究成果.在此背景下，本文将综述这些面向回归问题的降维算法，即实值多变量维数约简.本文将介绍与实值多变量维数约简密切相关的基本概念、算法、理论，并探讨一些潜在的研究方向.Abstract: As one of the classical problems in machine learning, dimension reduction is used for dealing with the curse of dimensionality, speeding up computational efficiency of the algorithm, and improving interpretability as well as visualizing high-dimensional data. Traditional dimension reduction algorithms such as principal component analysis (PCA) and linear discriminant analysis are mainly suitable for unlabeled data or classification data. When the response variables are univariate or multivariate continuous real-valued ones, however, such dimension reduction methods cannot guarantee the effective predictive performance of the reduced subspace. In the recent two decades, researchers have been devoted to studying this issue with different viewpoints, attaining many promising and systemic achievements. Under this background, we will survey the developments of real-valued multivariate dimension reduction in detail. We will also introduce its basic concepts, algorithms and theories, and discuss some potential research directions deserving investigating.1) 本文责任编委 朱军
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图 1 实值多变量维数约简算法分类
Fig. 1 Taxonomy of real-valued multivariate dimension reduction algorithms
表 1 基于互信息的降维算法对比
Table 1 Comparison of mutual information-based dimension reduction algorithms
目标函数 条件独立性 代表方法 解释 $\min_{{B}} {\rm E}_U \left[\mathbb{I}({Y|U};{V|U}) \right]$ ${Y}╨{V}|{U}$ KDR[32-33] 去掉$X$中与$Y$独立的一些特征, 也就是说这些特征对$Y$的预测不提供任何信息 $\max_{{B}} \mathbb{I}({Y};{U})$ ${Y}╨{X}|{U}$ LSDR[34-35]、MIDR[36] 从与$Y$相关的因素中寻找一个小的子空间或者子集, 不属于该子空间或者子集的特征, 虽然对$Y$的预测作出贡献, 但是在给定子空间或者子集后, 这部分信息不足以增加或改进对$Y$的预测能力 
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表 2 实值多变量维数约简算法总结
Table 2 Summary of real-valued multivariate dimension reduction algorithms
算法 响应变量 求解类型 分布假设 统计矩 模型选取 回归方式 #参数 参数列表 SIR[23] 一元 解析解 椭圆对称 一阶 无 逆回归 1 切片数$h$ SAVE[26] 一元 解析解 椭圆对称 二阶 无 逆回归 1 $h$ SIRII[49] 一元 解析解 椭圆对称 二阶 无 逆回归 1 $h$ pHd[25] 一元 解析解 正态分布 二阶 - 正回归 0 - DR[27] 一元 解析解 渐进正态 二阶 无 逆回归 1 $h$ PFC[28] 一元 解析解 模型假设 模型相关 无 逆回归 1 次数$r$ KDR[33] 多元 非凸优化 无 无穷 无 正回归 3 带宽$\sigma_x, \sigma_y$, $\epsilon$ LSDR[35] 多元 非凸优化 无 无穷 交叉验证 正回归 4 $\sigma_x, \sigma_y$, $\lambda$, $b$ MIDR[36] 多元 非凸优化 无 无穷 - 正回归 0 - HSIC[37] 多元 非凸优化 无 无穷 无 正回归 2 $\sigma_x, \sigma_y$ DCOV[38] 多元 非凸优化 无 二阶 - 正回归 0 - gKDR[40] 多元 解析解 无 无穷 无 正回归 3 $\sigma_x, \sigma_y, \epsilon$ LSGDR[41] 多元 解析解 无 无穷 交叉验证 正回归 $2D+1$ $\{\sigma_x^{(j)}, \lambda^{(j)}\}_{j=1}^D, \sigma_y$ KSIR[42] 一元 解析解 无 无穷 无 逆回归 2 $\sigma_x$, $h$ COIR[8] 多元 解析解 无 无穷 无 逆回归 3 $\sigma_x, \sigma_y, \epsilon$
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表 3 基于广义特征值求解的实值多变量维约简算法估计量
Table 3 Estimation of generalized eigen-decomposition-based multivariate dimension reduction algorithms
方法 (样本)估计量${M}$ SIR[23] ${\rm E}\Big[{\rm E}(Z|Y){\rm E}(Z|Y)^{\rm T}\Big] $ SAVE[26] ${\rm E}\Big[{I}-{\rm var}(Z|Y)\Big]^2$ SIRII[49] ${\rm E}\Big[{\rm var}(Z|Y)-{\rm E}({\rm var}(Z|Y))\Big]^2$ pHd[25] ${\rm E}\Big[(Y-E Y)ZZ^{\rm T}\Big]^2$ DR[27] $2{\rm E}\Big[E^2(ZZ^{\rm T}|Y)\Big]+ 2E^2\Big[{\rm E}(Z|Y){\rm E}(Z^{\rm T}|Y)\Big]+$ $ 2{\rm E}\Big[{\rm E}(Z^{\rm T}|Y){\rm E}(Z|Y)\Big]{\rm E}\Big[{\rm E}(Z|Y) {\rm E}(Z^{\rm T}|Y)\Big]\!-\!2{I}$ PFC[28] $\frac{1}{n}\widehat{{X}}^{\rm T}\widehat{{X}}$ gKDR[40] $ \big\langle \frac{\partial \kappa_x(\cdot, {\pmb x})}{\partial {\pmb x}}, \Sigma_{XX}^{-1}\Sigma_{XY}\Sigma_{YX}\Sigma_{XX}^{-1}\frac{\partial \kappa_x(\cdot, {\pmb x})}{\partial {\pmb x}}\big\rangle_{\mathcal{H}_x}$ LSGDR[41] $ {\rm E}({{\pmb g}}(y, {\pmb x}){{\pmb g}}(y, {\pmb x})^{\rm T})$ KSIR[42] ${\rm var}({\rm E}[{ \pmb \phi}({\pmb x})|y])$ COIR[8] ${\rm var}({\rm E}[{ \pmb \phi}({\pmb x})|y])$
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表 4 通过矩阵流形优化的降维算法的总结
Table 4 Summary of matrix manifold optimization-based dimension reduction algorithms
算法 目标函数$f({B})$ 欧式空间梯度$\nabla_{{B}} f$ 对应的权值矩阵${W}$ KDR[33] ${\rm{tr}}[{\overline{K}}_Y({\overline{K}}_U+n\epsilon {I}_n)^{-1}]$ $\frac{2}{\sigma^2}{X}{L}_{{\rm{KDR}}}{X}^{\rm T}{B}$ $\begin{array}{rl} {W}_{hrm{KDR}} &\!\!\!\!\!\!\!={H}{\Omega}{H}\odot{K}_U\\ {\Omega} &\!\!\!\!\!\!\!= {M}^{-1}{H}{K}_Y{H}{M}^{-1} \\ {M} &\!\!\!\!\!\!\!= {H}{K}_U{H}+n \epsilon {I} \end{array}$ LSDR[35] $\frac{1}{2}{\hat{\pmb{\alpha}}}^{\rm T}{\widehat{H}}{\hat{\pmb{\alpha}}}- \widehat{\pmb{h}}^{\rm T}\widehat{\pmb{\alpha}}$ 不存在解析形式梯度 无拉普拉斯形式 MIDR[36] $\begin{array}{ll}&\frac{d+1}{n(n-1)}\sum_{i\neq j}\log(\left\| {({\pmb x}_i- {\pmb x}_j){B}} \right\|^2+\left\| {y_i-y_j} \right\|^2)-\\ &\frac{d}{n(n-1)}\sum_{i\neq j}\log\left\| {({\pmb x}_i-{\pmb x}_j){B}} \right\| ^2\end{array}$ $-\frac{4}{n(n-1)} {X}{L}_{{\rm{MIDR}}}{X}^{\rm T}{B}$ $\begin{array}{rl}&{W}_{hrm{MIDR}}=\frac{d}{{D}_X\!\odot\!{D}_X}- \\ &\frac{d+1}{({D}_X\!\odot\!{D}_X\!+\!{D}_Y\!\odot\!{D}_Y)}\end{array}$ HSIC[37] $-\frac{1}{(n-1)^2}\textrm{tr}\big[{K}_U{H}{K}_Y {H}\big]$ $\frac{2}{\sigma^2(n-1)^2}{X}{L}_{{\rm{HSIC}}}{X}^{\rm T}{B}$ ${W}_{{\rm{HSIC}}} = {H}{K}_Y{H}\odot{K}_U$ DCOV[38] $-\frac{1}{n^2}{\rm{tr}}\big[{\overline{D}}_U{\overline{D}}_Y\big]$ $-\frac{2}{n^2}{X}{L}_{{\rm{DCOV}}}{X}^{\rm T}{B}$ ${W}_{{\rm{DCOV}}} = \frac{{H}{D}_Y{H}}{{D}_U}$
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