Density estimation on manifolds with boundary

Density estimation is a crucial component of many machine learning methods, and manifold learning in particular, where geometry is to be constructed from data alone. A significant practical limitation of the current density estimation literature is that methods have not been developed for manifolds with boundary, except in simple cases of linear manifolds where the location of the boundary is assumed to be known. We overcome this limitation by developing a density estimation method for manifolds with boundary that does not require any prior knowledge of the location of the boundary. To accomplish this we introduce statistics that provably estimate the distance and direction of the boundary, which allows us to apply a cut-and-normalize boundary correction. By combining multiple cut-and-normalize estimators we introduce a consistent kernel density estimator that has uniform bias, at interior and boundary points, on manifolds with boundary.