  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.
