The multi-index model with sparse dimension reduction matrix is a popular approach to circumvent the curse of dimensionality in a high-dimensional regression setting. Building on the single-index analysis by Alquier, P. & Biau, G. (Journal of Machine Learning Research 14 (2013) 243-280), we develop a PAC-Bayesian estimation method for a possibly misspecified multi-index model with unknown active dimension and an orthogonal dimension reduction matrix. Our main result is a non-asymptotic oracle inequality, which shows that the estimation method adapts to the active dimension of the model, the sparsity of the dimension reduction matrix and the regularity of the link function. Under a Sobolev regularity assumption on the link function the estimator achieves the minimax rate of convergence (up to a logarithmic factor) and no additional price is paid for the unknown active dimension.

翻译：多指标模型结合稀疏降维矩阵，是高维回归背景下规避维数灾难的常用方法。基于Alquier, P.与Biau, G.对单指标模型的分析（Journal of Machine Learning Research 14 (2013) 243-280），本文针对可能误设的、具有未知活跃维度及正交降维矩阵的多指标模型，提出一种PAC-贝叶斯估计方法。我们的主要结果是一条非渐近的oracle不等式，表明该估计方法能够自适应于模型的活跃维度、降维矩阵的稀疏性以及连接函数的正则性。在连接函数满足索伯列夫正则性假设的条件下，该估计量达到了极小极大收敛速率（至多相差一个对数因子），且无需为未知活跃维度付出额外代价。