On almost Linearity of Low Dimensional Projections from High Dimensional Data

By Peter Hall, Ker-Chau Li. 1993

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1. Hansheng Wang
2. Bing Li
3. Xiangrong Yin
4. Yingcun Xia
5. Liping Zhu
6. Lixing Zhu
7. Chenlei Leng
8. Heng Peng

Citations in the corpus

• Sufficient Forecasting Using Factor Models (2015, background): “The linearity condition is satisfied when the distribution of f t is elliptically symmetric (Eaton, 1983), and it is also asymptotically justified when the dimension of f t is large (Hall and Li, 1993).”
• Inverse Moment Methods for Sufficient Forecasting Using High-Dimensional Predictors (2016, unknown): “han K, the low-dimensional projection E(b0f tj˚ 0 1 f t; ;˚ 0 L f t) from a high-dimensional random vector f t will always approximate to a linear function under fairly general regularity conditions (Hall and Li, 1993), and the ellipticity of f t is no longer needed. Assumption (A2) can be restricti”
• Sufficient Dimension Reduction via Inverse Regression (2005, background): “Additionally, Hall and Li (1993) showed that as p increases with dim(SY”
• Graphics for regressions with a binary response (1996, background): “This condition is guaranteed when the predictors follow an elliptically contoured distribution (Eaton 1986) and may be expected to hold approximately when p is large (Hall and Li 1993).”
• An RKHS formulation of the inverse regression dimension-reduction problem (2009, background): “Then HR contains elements spanned by the column vectors of R, where 〈f,g〉HR = E(fT R−XXT R−g) = fT R−g(7) and −1X f = (R−f)T X, f ∈ HR,(8) R− being the Moore–Penrose generalized inverse of R. Li (1991) showed that Z ∈ L2X,e with probability 1, and it follows that Lg = E ( Z ⊗ HR Z ) g = E(〈Z,g〉HRZ) ”
• Model-Free Feature Screening for Ultrahigh Dimensional Data (2011, background): “Furthermore, Hall and Li (1993) showed that the linearity condition holds asymptotically if the number of predictors p diverges while the dimension K remains fixed.”
• Independent screening for single‐index hazard rate models with ultrahigh dimensional features (2011, background): “In view of Hall and Li (1993) which states that most low dimensional projections of high dimensional features are close to linear, Assumption 2 may not be unreasonable a priori even for general feature distributions when pn is large. Assumption 3 restricts the censoring mechanism to be partially ran”
• Asymptotics for pooled marginal slicing estimator based on SIRα approach (2005, methodology): “Using a Bayesian argument of Hall and Li [12], we can infer that (2) holds approximatively for many high dimensional data sets.”
• Fused Estimators of the Central Subspace in Sufficient Dimension Reduction (2014, background): “The problem of choosing the number of slices was first noticed when Li (1991) established SIR.”
• On Directional Regression for Dimension Reduction (2007, unknown): “Using the results of Diaconis and Freedman (1984) and Hall and Li (1993), it can be shown that as p → ∞, the (random) characteristic function of the conditional distribution of ηT Z|η converges in probability to a normal characteristic function.”
• Sliced Regression for Dimension Reduction (2008, background): “Moreover, Hall and Li (1993) showed that the linearity condition is still approximately satisfied when the dimension of the X is sufficiently large.”