This paper proposes a novel discriminative regression method, called adaptive locality preserving regression (ALPR) for classification. In particular, ALPR aims to learn a more flexible and discriminative projection that not only preserves the intrinsic structure of data, but also possesses the properties of feature selection and interpretability. To this end, we introduce a target learning technique to adaptively learn a more discriminative and flexible target matrix rather than the pre-defined strict zero-one label matrix for regression. Then a locality preserving constraint regularized by the adaptive learned weights is further introduced to guide the projection learning, which is beneficial to learn a more discriminative projection and avoid overfitting. Moreover, we replace the conventional `Frobenius norm' with the special l21 norm to constrain the projection, which enables the method to adaptively select the most important features from the original high-dimensional data for feature extraction. In this way, the negative influence of the redundant features and noises residing in the original data can be greatly eliminated. Besides, the proposed method has good interpretability for features owing to the row-sparsity property of the l21 norm. Extensive experiments conducted on the synthetic database with manifold structure and many real-world databases prove the effectiveness of the proposed method.

This paper describes a novel framework to jointly learn data-dependent label and locality-preserving projections. Given a set of data instances from multiple classes, the proposed approach can automatically learn which classes are more similar to each other, and construct discriminative features using both labeled and unlabeled data to map similar classes to similar locations in a lower dimensional space. In contrast to linear discriminant analysis (LDA) and its variants, which can only return c-1 features for a problem with c classes, the proposed approach can generate d features, where d is bounded only by the number of the input features. We describe and evaluate the new approach both theoretically and experimentally, and compare its performance with other state of the art methods.