Sparsity Preserving Discriminant Projections with Applications to Face Recognition

Dimensionality reduction is extremely important for understanding the intrinsic structure hidden in high-dimensional data. In recent years, sparse representation models have been widely used in dimensionality reduction. In this paper, a novel supervised learning method, called Sparsity Preserving Discriminant Projections (SPDP), is proposed. SPDP, which attempts to preserve the sparse representation structure of the data and maximize the between-class separability simultaneously, can be regarded as a combiner of manifold learning and sparse representation. Specifically, SPDP first creates a concatenated dictionary by classwise PCA decompositions and learns the sparse representation structure of each sample under the constructed dictionary using the least square method. Secondly, a local between-class separability function is defined to characterize the scatter of the samples in the different submanifolds. Then, SPDP integrates the learned sparse representation information with the local between-class relationship to construct a discriminant function. Finally, the proposed method is transformed into a generalized eigenvalue problem. Extensive experimental results on several popular face databases demonstrate the feasibility and effectiveness of the proposed approach

Manifold Elastic Net: A Unified Framework for Sparse Dimension Reduction

It is difficult to find the optimal sparse solution of a manifold learning based dimensionality reduction algorithm. The lasso or the elastic net penalized manifold learning based dimensionality reduction is not directly a lasso penalized least square problem and thus the least angle regression (LARS) (Efron et al. \cite{LARS}), one of the most popular algorithms in sparse learning, cannot be applied. Therefore, most current approaches take indirect ways or have strict settings, which can be inconvenient for applications. In this paper, we proposed the manifold elastic net or MEN for short. MEN incorporates the merits of both the manifold learning based dimensionality reduction and the sparse learning based dimensionality reduction. By using a series of equivalent transformations, we show MEN is equivalent to the lasso penalized least square problem and thus LARS is adopted to obtain the optimal sparse solution of MEN. In particular, MEN has the following advantages for subsequent classification: 1) the local geometry of samples is well preserved for low dimensional data representation, 2) both the margin maximization and the classification error minimization are considered for sparse projection calculation, 3) the projection matrix of MEN improves the parsimony in computation, 4) the elastic net penalty reduces the over-fitting problem, and 5) the projection matrix of MEN can be interpreted psychologically and physiologically. Experimental evidence on face recognition over various popular datasets suggests that MEN is superior to top level dimensionality reduction algorithms.Comment: 33 pages, 12 figure