  The knowledge that data lies close to a particular submanifold of the ambient
Euclidean space may be useful in a number of ways. For instance, one may want
to automatically mark any point far away from the submanifold as an outlier or
to use the geometry to come up with a better distance metric. Manifold learning
problems are often posed in a very high dimension, e.g. for spaces of images or
spaces of words. Today, with deep representation learning on the rise in areas
such as computer vision and natural language processing, many problems of this
kind may be transformed into problems of moderately high dimension, typically
of the order of hundreds. Motivated by this, we propose a manifold learning
technique suitable for moderately high dimension and large datasets. The
manifold is learned from the training data in the form of an intersection of
quadric hypersurfaces -- simple but expressive objects. At test time, this
manifold can be used to introduce a computationally efficient outlier score for
arbitrary new data points and to improve a given similarity metric by
incorporating the learned geometric structure into it.
