LMNN employs a large-margin framework to learn a generalized chi-squared distance for histogram data and obtains a significant improvement compared to standard histogram metrics and the state-of-the-art metric learning algorithms. Noh uses a convex optimization method to perform chi-squared metric learning with relaxation. Subsequently, EMDL formulates the ground metric learning as an optimization problem in which a ground distance matrix and a flow-network for the EMD are learned jointly based on a partial ordering of histogram distances. Considering that the ground distance, which is the unique variable of the EMD, should be chosen according to the problem at hand, Cuturi and Avis propose a ground metric learning algorithm to learn the ground metric adaptively by using the training data. Aiming at this problem, some researchers have attempted to learn a proper distance metric from histogram training data. However, distance metric is problem-specific and designing a good distance metric manually is extremely difficult. For the methods mentioned above, the determinations of metrics are all based on a priori knowledge of features or handcraft. TEMD uses a tangent vector to represent each global transformation. FastEMD adopts a robust thresholded ground distance and was shown to outperform the EMD in both accuracy and speed. Pele and Werman propose a different formulation of the EMD with a linear-time algorithm for nonnormalized histograms. EMD- uses the distance as the ground distance and significantly simplifies the original linear programming formulation of the EMD.
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In particular, for the cross-bin distance, most of the work mainly focuses on how to improve the EMD and hence many variants have been proposed. The Quadratic-Chi distances (QCS and QCN) take into account cross-bin relationships and meanwhile reduce the effect of large bins. Diffusion distance exploits the idea of diffusion process to define the difference between two histograms as a temperature field. propose the Earth Movers Distance (EMD), which is defined as the minimal cost that must be paid to transform one histogram into the other, by considering the cross-bin information. To mitigate these problems, many cross-bin distances have been proposed. These metrics, however, only account for the difference between the corresponding bins and are hence sensitive to distortions in visual descriptors as well as quantization effects. Since a histogram can be considered as a vector of probability, many metrics such as distance, chi-squared distance, and Kullback-Leibler (KL) divergence can be used directly. When the histogram representations are adopted, the choice of histogram distance metric has a great impact on the classification performance or recognition accuracy of the specific task. These make it an excellent representation method for performing classification and recognition of objects. As a result, the resulting histogram obtains some merits of the descriptors, for example, rotation-invariant, scale-invariant, and translation-invariant. For many computer vision tasks, each object of interest can be presented as a histogram by using visual descriptors, such as SIFT, SURF, GIST, and HOG. In particular, a histogram in the statistics is the frequency distribution of a set of specific measurements over discrete intervals. Histograms are frequently used tools in natural language processing and various computer vision tasks, including image retrieval, image classification, shape matching, and object recognition, to represent texture and color features or to characterize rich information in local/global regions of objects. Comparative studies with the state-of-the-art approaches on five real-world datasets verify the effectiveness of the proposed method. With the iterative projected gradient method for optimization, we naturally introduce the norm regularization into the proposed method for sparse metric learning. In our method, the margin of sample is first defined with respect to the nearest hits (nearest neighbors from the same class) and the nearest misses (nearest neighbors from the different classes), and then the simplex-preserving linear transformation is trained by maximizing the margin while minimizing the distance between each sample and its nearest hits. In this paper, we show how to learn a general form of chi-squared distance based on the nearest neighbor model. The chi-squared distance is a nonlinear metric and is widely used to compare histograms. Learning a proper distance metric for histogram data plays a crucial role in many computer vision tasks.