We present a new dimensionality reduction setting for a large family of real-world problems. Unlike traditional methods, the new setting aims to explicitly represent and learn an intrinsic structure from data in a high-dimensional space, which can greatly facilitate data visualization and scientific discovery in downstream analysis. We propose a new dimensionality-reduction framework that involves the learning of a mapping function that projects data points in the original high-dimensional space to latent points in a low-dimensional space that are then used directly to construct a graph. Local geometric information of the projected data is naturally captured by the constructed graph. As a showcase, we develop a new method to obtain a discriminative and compact feature representation for clustering problems. In contrast to assumptions used in traditional clustering methods, we assume that centers of clusters should be close to each other if they are connected in a learned graph, and other cluster centers should be distant. Extensive experiments are performed that demonstrate that the proposed method is able to obtain discriminative feature representations yielding superior clustering performance, and correctly recover the intrinsic structures of various real-world datasets including curves, hierarchies and a cancer progression path.