High-dimensional crowdsourced data collected from numerous users produces rich knowledge about our society; however, it also brings unprecedented privacy threats to the participants. Local differential privacy (LDP), a variant of differential privacy, is recently proposed as a state-of-the-art privacy notion. Unfortunately, achieving LDP on high-dimensional crowdsourced data publication raises great challenges in terms of both computational efficiency and data utility. To this end, based on the expectation maximization (EM) algorithm and Lasso regression, we first propose efficient multi-dimensional joint distribution estimation algorithms with LDP. Then, we develop a local differentially private high-dimensional data publication algorithm (LoPub) by taking advantage of our distribution estimation techniques. In particular, correlations among multiple attributes are identified to reduce the dimensionality of crowdsourced data, thus speeding up the distribution learning process and achieving high data utility. Extensive experiments on real-world datasets demonstrate that our multivariate distribution estimation scheme significantly outperforms existing estimation schemes in terms of both communication overhead and estimation speed. Moreover, LoPub can keep, on average, 80% and 60% accuracy over the released datasets in terms of support vector machine and random forest classification, respectively.