We consider regression problems where the response is a smooth but non-linear function of a $k$-dimensional projection of $p$ normally-distributed covariates, contaminated with additive Gaussian noise. The goal is to recover the range of the $k$-dimensional projection, i.e., the index space. This model is called the multi-index model, and the $k=1$ case is called the single-index model. For the single-index model, we characterize the population landscape of a natural semi-parametric maximum likelihood objective in terms of the link function and prove that it has no spurious local minima. We also propose and analyze an efficient iterative procedure that recovers the index space up to error $epsilon$ using a sample size $ilde{O}(p^{O(R^2/mu)} + p/epsilon^2)$, where $R$ and $mu$, respectively, parameterize the smoothness of the link function and the signal strength. When a multi-index model is incorrectly specified as a single-index model, we prove that essentially the same procedure, with sample size $ilde{O}(p^{O(kR^2/mu)} + p/epsilon^2)$, returns a vector that is $epsilon$-close to being completely in the index space.
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机译：我们考虑回归问题，其中响应是$ p $正态分布协变量的$ k $维投影的平滑但非线性函数，并被加性高斯噪声污染。目标是恢复$ k $维投影的范围，即索引空间。此模型称为多索引模型，$ k = 1 $的情况称为单索引模型。对于单指标模型，我们通过链接函数来表征自然半参数最大似然目标的总体格局，并证明它没有虚假的局部最小值。我们还提出并分析了一个有效的迭代过程，该过程使用样本大小$ tilde {O}（p ^ {O（R ^ 2 / mu）} + p / epsilon将索引空间恢复到错误$ epsilon $。 ^ 2）$，其中$ R $和$ mu $分别参数化链接函数的平滑度和信号强度。当多索引模型被错误地指定为单索引模型时，我们证明样本量为$ tilde {O}（p ^ {O（kR ^ 2 / mu）} + p / epsilon ^ 2）$，返回为$ epsilon $-接近于完全位于索引空间中的向量。
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