  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 its geodesic distance to measure similarity between points. Classical
problems for manifold learning are often posed in a very high dimension, e.g.
for spaces of images or spaces of representations 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 an
outlier score for arbitrary new points and to improve a given similarity metric
by incorporating learned geometric structure into it.
