Scalable interpretable learning for multi-response error-in-variables regression

Corrupted data sets containing noisy or missing observations are prevalent in various contemporary applications such as economics, finance and bioinformatics. Despite the recent methodological and algorithmic advances in high-dimensional multi-response regression, how to achieve scalable and interpretable estimation under contaminated covariates is unclear. In this paper, we develop a new methodology called convex conditioned sequential sparse learning (COSS) for error-in-variables multi-response regression under both additive measurement errors and random missing data. It combines the strengths of the recently developed sequential sparse factor regression and the nearest positive semi-definite matrix projection, thus enjoying stepwise convexity and scalability in large-scale association analyses. Comprehensive theoretical guarantees are provided and we demonstrate the effectiveness of the proposed methodology through numerical studies.