Georgia Institute of Technology College of Sciences

Many data set in image analysis and signal processing is in a high-dimensional space but exhibit low-dimensional structures. For example, data can be modeled as point clouds in a high-dimensional space but are concentrated on a low-dimensional set (or manifold in particular). Our goal is to estimate functions on the low-dimensional manifold from finite samples of data, for statistical inference and prediction. This talk introduces approximation theories of neural networks for functions supported on a low-dimensional manifold. When the function is estimated from finite samples, we give an estimate of the mean squared error for the approximation of these functions. The convergence rate depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. These results demonstrate that neural networks are adaptive to low-dimensional geometric structures of data.