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

Quadric Hypersurface Intersection for Manifold Learning in Feature Space