Density Estimation via Discrepancy

Since these data are typically sampled from multi-modal distributions, a natural choice would be using nonparametric density estimation methods. Classic empirical distribution (ED) and kernel density estimation (KDE) play an important role in nonparametric density estimation. Besides their long noticed drawbacks (e.g., ED is noncontinuous; KDE is sensitive to the choice of bandwidth and scales poorly in high dimensions), they are not good summarization tools in dealing with data with high dimension and large size, e.g., evaluating them involves each data point and their functional forms provide little direct information of the "landscape" of the distribution. In this paper, we consider domain partition based approach for density estimation. The use of domain partition dates back to histogram, which is still an ubiquitous tool in data analysis today; however, its non-scalability in high dimensions limits its applications. Motivated by the usefulness of histogram and the attempts to adapt it for multivariate cases, we propose a novel nonparametric density estimation method.