Regression with a spherical response is challenging due to the absence of linear structure, making standard regression models inadequate. Existing methods, mainly parametric, lack the flexibility to capture the complex relationship induced by spherical curvature, while methods based on techniques from Riemannian geometry often suffer from computational difficulties. The non-Euclidean structure further complicates robust estimation, with very limited work addressing this issue, despite the common presence of outliers in directional data. This article introduces a new semi-parametric approach, the extrinsic single-index model (ESIM) and its robust estimation, to address these limitations. We establish large-sample properties of the proposed estimator with a wide range of loss functions and assess their robustness using the influence function and standardized influence function. Specifically, we focus on the robustness of the exponential squared loss (ESL), demonstrating comparable efficiency and superior robustness over least squares loss under high concentration. We also examine how the tuning parameter for the ESL balances efficiency and robustness, providing guidance on its optimal choice. The computational efficiency and robustness of our methods are further illustrated via simulations and applications to geochemical compositional data.

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