Learning Mahalanobis metric spaces is an important problem that has found numerous applications. Several algorithms have been designed for this problem, including Information Theoretic Metric Learning (ITML) by [ 2007] and Large Margin Nearest Neighbor (LMNN) classification by [Weinberger and Saul 2009]. We consider a formulation of Mahalanobis metric learning as an optimization problem, where the objective is to minimize the number of violated similarity dissimilarity constraints. We show that for any fixed ambient dimension, there exists a fully polynomial-time approximation scheme (FPTAS) with nearlylinear running time. This result is obtained using tools from the theory of linear programming in low dimensions. We also discuss improvements of the algorithm in practice, and present experimental results on synthetic and real-world data sets.
Several algorithms for learning Mahalanobis metric spaces have been proposed. Notable examples include the SDP based algorithm of @cite_14 , the algorithm of Globerson and Roweis for the fully supervised setting @cite_13 , Information Theoretic Metric Learning ( ) by @cite_2 , which casts the problem as a particular optimization minimizing LogDet divergence, as well as Large Margin Nearest Neighbor ( ) by @cite_10 , which attempts to learn a metric geared towards optimizing @math -NN classification. We refer the reader to the surveys @cite_11 @cite_4 for a detailed discussion of previous work. Our algorithm differs from previous approaches in that it seeks to directly minimize the number of violated pairwise distance constraints, which is a highly non-convex objective, without resorting to a convex relaxation of the corresponding optimization problem.
@cite_13 @cite_14 @cite_4 @cite_2 @cite_10 @cite_11
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We present an algorithm for learning a quadratic Gaussian metric (Mahalanobis distance) for use in classification tasks. Our method relies on the simple geometric intuition that a good metric is one under which points in the same class are simultaneously near each other and far from points in the other classes. We construct a convex optimization problem whose solution generates such a metric by trying to collapse all examples in the same class to a single point and push examples in other classes infinitely far away. We show that when the metric we learn is used in simple classifiers, it yields substantial improvements over standard alternatives on a variety of problems. We also discuss how the learned metric may be used to obtain a compact low dimensional feature representation of the original input space, allowing more efficient classification with very little reduction in performance. Many algorithms rely critically on being given a good metric over their inputs. For instance, data can often be clustered in many "plausible" ways, and if a clustering algorithm such as K-means initially fails to find one that is meaningful to a user, the only recourse may be for the user to manually tweak the metric until sufficiently good clusters are found. For these and other applications requiring good metrics, it is desirable that we provide a more systematic way for users to indicate what they consider "similar." For instance, we may ask them to provide examples. In this paper, we present an algorithm that, given examples of similar (and, if desired, dissimilar) pairs of points in ℝn, learns a distance metric over ℝn that respects these relationships. Our method is based on posing metric learning as a convex optimization problem, which allows us to give efficient, local-optima-free algorithms. We also demonstrate empirically that the learned metrics can be used to significantly improve clustering performance. Abstract Distance metric learning has been a hot research spot recently due to its high effectiveness and efficiency in improving the performance of distance related methods, such as k nearest neighbors ( k NN). Metric learning aims to learn a data-dependent metric to make intra-class distance smaller and inter-class larger. A large number of methods have been proposed for various applications and a survey to evaluate and compare these methods is imperative. The existing surveys just analyze the algorithms theoretically or compare them experimentally with a narrow time scope. Therefore, the paper reviews classical and influential methods that were proposed between 2003 and 2017 and presents a taxonomy based on the most distinct character of each method. All the methods are categorized into five classes, including pairwise cost, probabilistic framework, boost-like approaches, advantageous variants and specific applications. A comprehensive experimental study is made to compare all the selected methods, exploring the ability in improving accuracy, the relation between distance change and accuracy, the relation between accuracy and k NN neighbor size. We show how to learn a Mahanalobis distance metric for k-nearest neighbor (kNN) classification by semidefinite programming. The metric is trained with the goal that the k-nearest neighbors always belong to the same class while examples from different classes are separated by a large margin. On seven data sets of varying size and difficulty, we find that metrics trained in this way lead to significant improvements in kNN classification—for example, achieving a test error rate of 1.3 on the MNIST handwritten digits. As in support vector machines (SVMs), the learning problem reduces to a convex optimization based on the hinge loss. Unlike learning in SVMs, however, our framework requires no modification or extension for problems in multiway (as opposed to binary) classification. The metric learning problem is concerned with learning a distance function tuned to a particular task, and has been shown to be useful when used in conjunction with nearest-neighbor methods and other techniques that rely on distances or similarities. This survey presents an overview of existing research in metric learning, including recent progress on scaling to high-dimensional feature spaces and to data sets with an extremely large number of data points. A goal of the survey is to present as unified as possible a framework under which existing research on metric learning can be cast. The first part of the survey focuses on linear metric learning approaches, mainly concentrating on the class of Mahalanobis distance learning methods. We then discuss nonlinear metric learning approaches, focusing on the connections between the nonlinear and linear approaches. Finally, we discuss extensions of metric learning, as well as applications to a variety of problems in computer vision, text analysis, program analysis, and multimedia. Full text available at: http: dx.doi.org 10.1561 2200000019