Spherical Regression Under Mismatch Corruption With Application to Automated Knowledge Translation

Motivated by a series of applications in data integration, language translation, bioinformatics, and computer vision, we consider spherical regression with two sets of unit-length vectors when the data are corrupted by a small fraction of mismatch in the response-predictor pairs. We propose a three-step algorithm in which we initialize the parameters by solving an orthogonal Procrustes problem to estimate a translation matrix (Formula presented.) ignoring the mismatch. We then estimate a mapping matrix aiming to correct the mismatch using hard-thresholding to induce sparsity, while incorporating potential group information. We eventually obtain a refined estimate for (Formula presented.) by removing the estimated mismatched pairs. We derive the error bound for the initial estimate of (Formula presented.) in both fixed and high-dimensional setting. We demonstrate that the refined estimate of (Formula presented.) achieves an error rate that is as good as if no mismatch is present. We show that our mapping recovery method not only correctly distinguishes one-to-one and one-to-many correspondences, but also consistently identifies the matched pairs and estimates the weight vector for combined correspondence. We examine the finite sample performance of the proposed method via extensive simulation studies, and with application to the unsupervised translation of medical codes using electronic health records data. Supplementary materials for this article are available online.