A parameter-free framework for general supervised subspace learning

Supervised subspace learning techniques have been extensively studied in biometrics literature; however, there is little work dedicated to: 1) how to automatically determine the subspace dimension in the context of supervised learning, and 2) how to explicitly guarantee the classification performance on a training set. In this paper, by following our previous work on unified subspace learning framework in our earlier work, we present a general framework, called parameter-free graph embedding (PFGE) to solve the above two problems by posing a general supervised subspace learning task as a semidefinite programming problem. The semipositive feature Gram matrix, namely the product of the transformation matrix and its transpose, is derived by optimizing a trace difference form of an objective function extended from that in our earlier work with the constraints that guarantee the class homogeneity within the neighborhood of each datum. Then, the subspace dimension and the feature weights are simultaneously obtained via the singular value decomposition of the feature Gram matrix. In addition, to alleviate the computational complexity, the Kronecker product approximation of the feature Gram matrix is proposed by taking advantage of the essential matrix form of image pixels. The experiments on simulated data and real-world data demonstrate the capability of the new PFGE framework in estimating the subspace dimension for supervised learning as well as the superiority in classification performance over traditional algorithms for subspace learning.