Subspace Regression: Predicting a Subspace from one Sample

Subspace methods have been extensively used to solve a variety of problems in computer vision including object detection, recognition, and tracking. Typically the subspaces are learned from a training set that contains different configurations or states of a particular object (e.g., variations on shape or appearance). However, in some situations it is not possible to have access to data with multiple configurations of the object. An example is the problem of building a person-specific subspace of the pose variation by having only the frontal face. In this paper we propose methods to predict a subspace from only one sample. Predicting a subspace from only one sample is a challenging problem for several reasons: (i) it involves a mapping in very high-dimensional spaces, and often there are far fewer training samples than the number of features, (ii) it is unclear how to parameterize the subspace. This paper proposes four methods that learn a mapping from one sample to a subspace: Individual Mapping on Images, Direct Mapping to Subspaces, Regression on Subspaces, and Direct Subspace Alignment. We show the validity of our approaches in predicting a person-specific face subspace of pose or illumination and applications to face tracking. To the best of our knowledge this is the first paper that addresses the problem of learning a subspace directly from one sample.