Eigenvector-Based Distance Metric Learning for Image Classification and Retrieval

Distance metric learning has been widely studied in multifarious research fields. The mainstream approaches learn a Mahalanobis metric or learn a linear transformation. Recent related works propose learning a linear combination of base vectors to approximate the metric. In this way, fewer variables need to be determined, which is efficient when facing high-dimensional data. Nevertheless, such works obtain base vectors using additional data from related domains or randomly generate base vectors. However, obtaining base vectors from related domains requires extra time and additional data, and random vectors introduce randomness into the learning process, which requires sufficient random vectors to ensure the stability of the algorithm. Moreover, the random vectors cannot capture the rich information of the training data, leading to a degradation in performance. Considering these drawbacks, we propose a novel distance metric learning approach by introducing base vectors explicitly learned from training data. Given a specific task, we can make a sparse approximation of its objective function using the top eigenvalues and corresponding eigenvectors of a predefined integral operator on the reproducing kernel Hilbert space. Because the process of generating eigenvectors simply refers to the training data of the considered task, our proposed method does not require additional data and can reflect the intrinsic information of the input features. Furthermore, the explicitly learned eigenvectors do not result in randomness, and we can extend our method to any kernel space without changing the objective function. We only need to learn the coefficients of these eigenvectors, and the only hyperparameter that we need to determine is the number of eigenvectors that we utilize. Additionally, an optimization algorithm is proposed to efficiently solve this problem. Extensive experiments conducted on several datasets demonstrate the effectiveness of our proposed method.