Abstract:   Privacy-preserving data analysis has been put on a firm mathematical foundation since the introduction of differential privacy (DP) in 2006. This privacy definition, however, has some well-known weaknesses: notably, it does not tightly handle composition. This weakness has inspired several recent relaxations of differential privacy based on the Renyi divergences. We propose an alternative…

Abstract: We introduce the diffusion K-means clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion K-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of…

Abstract: We introduce the jackknife+, a novel method for constructing predictive confidence intervals that is robust to the distribution of the data. The jackknife+ modifies the well-known jackknife (leaveoneout cross-validation) to account for the variability in the fitted regression function when we subsample the training data. Assuming exchangeable training samples, we prove that the jackknife+…

Abstract: Properties of random projections of high-dimensional probability measures are of interest in a variety of fields, including asymptotic convex geometry, and potential applications to high-dimensional statistics and data analysis.   A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections.   While fluctuations of these projections have been…