Quadric Hypersurface Intersection for Manifold Learning in Feature Space
This work addresses manifold learning challenges in domains like computer vision and NLP, where deep representation learning reduces dimensions to hundreds, but it appears incremental as it builds on existing geometric approaches.
The authors tackled the problem of manifold learning in moderately high-dimensional feature spaces by proposing a technique that models the manifold as an intersection of quadric hypersurfaces, enabling efficient outlier detection and similarity metric improvement for large datasets.
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.