Algorithms for Envelope Estimation

Envelopes were recently proposed as methods for reducing estimative variation in multivariate linear regression. Estimation of an envelope usually involves optimization over Grassmann manifolds. We propose a fast and widely applicable one-dimensional (1D) algorithm for estimating an envelope in general. We reveal an important structural property of envelopes that facilitates our algorithm, and we prove both Fisher consistency and n-consistency of the algorithm.包络（Envelope）方法作为降低多元线性回归估计变异的有效手段，近期被学界提出。包络估计通常需在格拉斯曼流形（Grassmann manifolds）上开展优化运算。针对一般场景下的包络估计问题，我们提出了一种快速且普适性优异的一维（1D）算法。我们揭示了包络的一项关键结构性质，该性质可简化本算法的实现流程，并证明了该算法的费希尔相合性（Fisher consistency）与n相合性（n-consistency）。