Data in high dimensional Euclidean space may often be described with fewer parameters than the ambient space, perhaps lying on or near a submanifold of lower intrinsic dimension. In part one of this talk, we will discuss a new method for estimating this intrinsic dimension from the data: a version of local PCA which is calibrated on quadratic embeddings which may better approximate a manifold at larger scales. The second part of the talk will focus on a technique of random embeddings to reduce ambient dimension. The topics of this talk are joint work with Anna Gilbert.