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We consider a high-dimensional monotone single index model (hdSIM), which isa semiparametric extension of a high-dimensional generalize linear model(hdGLM), where the link function is unknown, but constrained with monotone andnon-decreasing shape. We develop a scalable projection-based iterativeapproach, the "Sparse Orthogonal Descent Single-Index Model" (SOD-SIM), whichalternates between sparse-thresholded orthogonalized "gradient-like" steps andisotonic regression steps to recover the coefficient vector. Our maincontribution is that we provide finite sample estimation bounds for both thecoefficient vector and the link function in high-dimensional settings undervery mild assumptions on the design matrix $\mathbf{X}$, the error term$\epsilon$, and their dependence. The convergence rate for the link functionmatched the low-dimensional isotonic regression minimax rate up to somepoly-log terms ($n^{-1/3}$). The convergence rate for the coefficients is also$n^{-1/3}$ up to some poly-log terms. This method can be applied to many realdata problems, including GLMs with misspecified link, classification withmislabeled data, and classification with positive-unlabeled (PU) data. We studythe performance of this method via both numerical studies and also anapplication on a rocker protein sequence data.

Convergence guarantee for the sparse monotone single index model