As every day 2.5 quintillion bytes of data are generated, the era of Big Data is undoubtedly upon us. Nonetheless, a significant percentage of the data accrued can be omitted while maintaining a certain quality of statistical inference with a limited computational budget. In this context, estimating adaptively high-dimensional signals from massive data observed sequentially is challenging but equally important in practice. The present paper deals with this challenge based on a novel approach that leverages interval censoring for data reduction. An online maximum likelihood, least mean-square (LMS)-type algorithm, and an online support vector regression algorithm are developed for censored data. The proposed algorithms entail simple, low-complexity, closed-form updates, and have provably bounded regret. Simulated tests corroborate their efficacy.