A number of machine learning algorithms are using a metric, or a distance, in order to compare individuals. The Euclidean distance is usually employed, but it may be more efficient to learn a parametric distance such as Mahalanobis metric. Learning such a metric is a hot topic since more than ten years now, and a number of methods have been proposed to efficiently learn it. However, the nature of the problem makes it quite difficult for large scale data, as well as data for which classes overlap. This paper presents a simple way of improving accuracy and scalability of any iterative metric learning algorithm, where constraints are obtained prior to the algorithm. The proposed approach relies on a loss-dependent weighted selection of constraints that are used for learning the metric. Using the corresponding dedicated loss function, the method clearly allows to obtain better results than state-of-the-art methods, both in terms of accuracy and time complexity. Some experimental results on real world, and potentially large, datasets are demonstrating the effectiveness of our proposition.
Large Margin Nearest Neighbors (LMNN, @cite_30 ) is one of the early attempts to learn a Mahalanobis distance metric as a convex optimization problem over the set of PSD matrices. The loss function is composed of the linear combination of two terms @math and @math . The first term aims at penalizing large distances between an observation and other observations sharing the same label, while the second term objective is to penalize small distances between observations from different classes. The loss function is then casted as a semidefinite program. Note that there no regularization term in the objective function so that LMNN is prone to overfitting. Another well known problem, related to the proposed approach, is that the selection of neighbors is initially made using Euclidean distance, which may not be adapted.
@cite_30
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The accuracy of k-nearest neighbor (kNN) classification depends significantly on the metric used to compute distances between different examples. In this paper, we show how to learn a Mahalanobis distance metric for kNN classification from labeled examples. The Mahalanobis metric can equivalently be viewed as a global linear transformation of the input space that precedes kNN classification using Euclidean distances. In our approach, 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. As in support vector machines (SVMs), the margin criterion leads to a convex optimization based on the hinge loss. Unlike learning in SVMs, however, our approach requires no modification or extension for problems in multiway (as opposed to binary) classification. In our framework, the Mahalanobis distance metric is obtained as the solution to a semidefinite program. On several data sets of varying size and difficulty, we find that metrics trained in this way lead to significant improvements in kNN classification. Sometimes these results can be further improved by clustering the training examples and learning an individual metric within each cluster. We show how to learn and combine these local metrics in a globally integrated manner. In this paper, we present an information-theoretic approach to learning a Mahalanobis distance function. We formulate the problem as that of minimizing the differential relative entropy between two multivariate Gaussians under constraints on the distance function. We express this problem as a particular Bregman optimization problem---that of minimizing the LogDet divergence subject to linear constraints. Our resulting algorithm has several advantages over existing methods. First, our method can handle a wide variety of constraints and can optionally incorporate a prior on the distance function. Second, it is fast and scalable. Unlike most existing methods, no eigenvalue computations or semi-definite programming are required. We also present an online version and derive regret bounds for the resulting algorithm. Finally, we evaluate our method on a recent error reporting system for software called Clarify, in the context of metric learning for nearest neighbor classification, as well as on standard data sets. This paper presents a new method for viewpoint invariant pedestrian recognition problem. We use a metric learning framework to obtain a robust metric for large margin nearest neighbor classification with rejection (i.e., classifier will return no matches if all neighbors are beyond a certain distance). The rejection condition necessitates the use of a uniform threshold for a maximum allowed distance for deeming a pair of images a match. In order to handle the rejection case, we propose a novel cost similar to the Large Margin Nearest Neighbor (LMNN) method and call our approach Large Margin Nearest Neighbor with Rejection (LMNN-R). Our method is able to achieve significant improvement over previously reported results on the standard Viewpoint Invariant Pedestrian Recognition (VIPeR [1]) dataset. 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. 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. Metric learning has been shown to be highly effective to improve the performance of nearest neighbor classification. In this paper, we address the problem of metric learning for Symmetric Positive Definite (SPD) matrices such as covariance matrices, which arise in many real-world applications. Naively using standard Mahalanobis metric learning methods under the Euclidean geometry for SPD matrices is not appropriate, because the difference of SPD matrices can be a non-SPD matrix and thus the obtained solution can be uninterpretable. To cope with this problem, we propose to use a properly parameterized LogEuclidean distance and optimize the metric with respect to kernel-target alignment, which is a supervised criterion for kernel learning. Then the resulting non-trivial optimization problem is solved by utilizing the Riemannian geometry. Finally, we experimentally demonstrate the usefulness of our LogEuclidean metric learning algorithm on real-world classification tasks for EEG signals and texture patches. We address the problem of learning distance metrics using side-information in the form of groups of "similar" points. We propose to use the RCA algorithm, which is a simple and efficient algorithm for learning a full ranked Mahalanobis metric (, 2002). We first show that RCA obtains the solution to an interesting optimization problem, founded on an information theoretic basis. If the Mahalanobis matrix is allowed to be singular, we show that Fisher's linear discriminant followed by RCA is the optimal dimensionality reduction algorithm under the same criterion. We then show how this optimization problem is related to the criterion optimized by another recent algorithm for metric learning (, 2002), which uses the same kind of side information. We empirically demonstrate that learning a distance metric using the RCA algorithm significantly improves clustering performance, similarly to the alternative algorithm. Since the RCA algorithm is much more efficient and cost effective than the alternative, as it only uses closed form expressions of the data, it seems like a preferable choice for the learning of full rank Mahalanobis distances. The learning of appropriate distance metrics is a critical problem in image classification and retrieval. In this work, we propose a boosting-based technique, termed BOOSTMETRIC, for learning a Mahalanobis distance metric. One of the primary difficulties in learning such a metric is to ensure that the Mahalanobis matrix remains positive semidefinite. Semidefinite programming is sometimes used to enforce this constraint, but does not scale well. BOOSTMETRIC is instead based on a key observation that any positive semidefinite matrix can be decomposed into a linear positive combination of trace-one rank-one matrices. BOOSTMETRIC thus uses rank-one positive semidefinite matrices as weak learners within an efficient and scalable boosting-based learning process. The resulting method is easy to implement, does not require tuning, and can accommodate various types of constraints. Experiments on various datasets show that the proposed algorithm compares favorably to those state-of-the-art methods in terms of classification accuracy and running time. We are interested in supervised metric learning of Mahalanobis like distances. Existing approaches mainly focus on learning a new distance using similarity and dissimilarity constraints between examples. In this paper, instead of bringing closer examples of the same class and pushing far away examples of different classes we propose to move the examples with respect to virtual points. Hence, each example is brought closer to a a priori defined virtual point reducing the number of constraints to satisfy. We show that our approach admits a closed form solution which can be kernelized. We provide a theoretical analysis showing the consistency of the approach and establishing some links with other classical metric learning methods. Furthermore we propose an efficient solution to the difficult problem of selecting virtual points based in part on recent works in optimal transport. Lastly, we evaluate our approach on several state of the art datasets. We introduce a new probabilistic proximity search algorithm for range and A"-nearest neighbor (A"-NN) searching in both coordinate and metric spaces. Although there exist solutions for these problems, they boil down to a linear scan when the space is intrinsically high dimensional, as is the case in many pattern recognition tasks. This, for example, renders the A"-NN approach to classification rather slow in large databases. Our novel idea is to predict closeness between elements according to how they order their distances toward a distinguished set of anchor objects. Each element in the space sorts the anchor objects from closest to farthest to it and the similarity between orders turns out to be an excellent predictor of the closeness between the corresponding elements. We present extensive experiments comparing our method against state-of-the-art exact and approximate techniques, both in synthetic and real, metric and nonmetric databases, measuring both CPU time and distance computations. The experiments demonstrate that our technique almost always improves upon the performance of alternative techniques, in some cases by a wide margin. Support Vector Machines, SVMs, and the Large Margin Nearest Neighbor algorithm, LMNN, are two very popular learning algorithms with quite different learning biases. In this paper we bring them into a unified view and show that they have a much stronger relation than what is commonly thought. We analyze SVMs from a metric learning perspective and cast them as a metric learning problem, a view which helps us uncover the relations of the two algorithms. We show that LMNN can be seen as learning a set of local SVM-like models in a quadratic space. Along the way and inspired by the metric-based interpretation of SVM s we derive a novel variant of SVMs, epsilon-SVM, to which LMNN is even more similar. We give a unified view of LMNN and the different SVM variants. Finally we provide some preliminary experiments on a number of benchmark datasets in which show that epsilon-SVM compares favorably both with respect to LMNN and SVM. We consider unsupervised partitioning problems based explicitly or implicitly on the minimization of Euclidean distortions, such as clustering, image or video segmentation, and other change-point detection problems. We emphasize on cases with specific structure, which include many practical situations ranging from mean-based change-point detection to image segmentation problems. We aim at learning a Mahalanobis metric for these unsupervised problems, leading to feature weighting and or selection. This is done in a supervised way by assuming the availability of several (partially) labeled datasets that share the same metric. We cast the metric learning problem as a large-margin structured prediction problem, with proper definition of regularizers and losses, leading to a convex optimization problem which can be solved efficiently. Our experiments show how learning the metric can significantly improve performance on bioinformatics, video or image segmentation problems. We introduce a method that enables scalable image search for learned metrics. Given pairwise similarity and dissimilarity constraints between some images, we learn a Mahalanobis distance function that captures the imagespsila underlying relationships well. To allow sub-linear time similarity search under the learned metric, we show how to encode the learned metric parameterization into randomized locality-sensitive hash functions. We further formulate an indirect solution that enables metric learning and hashing for vector spaces whose high dimensionality make it infeasible to learn an explicit weighting over the feature dimensions. We demonstrate the approach applied to a variety of image datasets. Our learned metrics improve accuracy relative to commonly-used metric baselines, while our hashing construction enables efficient indexing with learned distances and very large databases. We address the problem of metric learning for multi-view data, namely the construction of embedding projections from data in different representations into a shared feature space, such that the Euclidean distance in this space provides a meaningful within-view as well as between-view similarity. Our motivation stems from the problem of cross-media retrieval tasks, where the availability of a joint Euclidean distance function is a prerequisite to allow fast, in particular hashing-based, nearest neighbor queries. We formulate an objective function that expresses the intuitive concept that matching samples are mapped closely together in the output space, whereas non-matching samples are pushed apart, no matter in which view they are available. The resulting optimization problem is not convex, but it can be decomposed explicitly into a convex and a concave part, thereby allowing efficient optimization using the convex-concave procedure. Experiments on an image retrieval task show that nearest-neighbor based cross-view retrieval is indeed possible, and the proposed technique improves the retrieval accuracy over baseline techniques. The distance metric is the corner stone of nearest neighbor (NN)-based methods, and therefore, of nearest prototype (NP) algorithms. That is because they classify depending on the similarity of the data. When the data is characterized by a set of features which may contribute to the classification task in different levels, feature weighting or selection is required, sometimes in a local sense. However, local weighting is typically restricted to NN approaches. In this paper, we introduce local feature weighting (LFW) in NP classification. LFW provides each prototype its own weight vector, opposite to typical global weighting methods found in the NP literature, where all the prototypes share the same one. Providing each prototype its own weight vector has a novel effect in the borders of the Voronoi regions generated: They become nonlinear. We have integrated LFW with a previously developed evolutionary nearest prototype classifier (ENPC). The experiments performed both in artificial and real data sets demonstrate that the resulting algorithm that we call LFW in nearest prototype classification (LFW-NPC) avoids overfitting on training data in domains where the features may have different contribution to the classification task in different areas of the feature space. This generalization capability is also reflected in automatically obtaining an accurate and reduced set of prototypes. In this paper we propose a novel method for learning a Mahalanobis distance measure to be used in the KNN classification algorithm. The algorithm directly maximizes a stochastic variant of the leave-one-out KNN score on the training set. It can also learn a low-dimensional linear embedding of labeled data that can be used for data visualization and fast classification. Unlike other methods, our classification model is non-parametric, making no assumptions about the shape of the class distributions or the boundaries between them. The performance of the method is demonstrated on several data sets, both for metric learning and linear dimensionality reduction. This paper presents a relational framework for studying properties of labeled data points related to proximity and labeling information in order to improve the performance of the 1NN rule. Specifically, the class conditional nearest neighbor (ccnn) relation over pairs of points in a labeled training set is introduced. For a given class label c, this relation associates to each point a its nearest neighbor computed among only those points with class label c (excluded a). A characterization of ccnn in terms of two graphs is given. These graphs are used for defining a novel scoring function over instances by means of an information-theoretic divergence measure applied to the degree distributions of these graphs. The scoring function is employed to develop an effective large margin instance selection method, which is empirically demonstrated to improve storage and accuracy performance of the 1NN rule on artificial and real-life data sets. In this paper, we raise important issues on scalability and the required degree of supervision of existing Mahalanobis metric learning methods. Often rather tedious optimization procedures are applied that become computationally intractable on a large scale. Further, if one considers the constantly growing amount of data it is often infeasible to specify fully supervised labels for all data points. Instead, it is easier to specify labels in form of equivalence constraints. We introduce a simple though effective strategy to learn a distance metric from equivalence constraints, based on a statistical inference perspective. In contrast to existing methods we do not rely on complex optimization problems requiring computationally expensive iterations. Hence, our method is orders of magnitudes faster than comparable methods. Results on a variety of challenging benchmarks with rather diverse nature demonstrate the power of our method. These include faces in unconstrained environments, matching before unseen object instances and person re-identification across spatially disjoint cameras. In the latter two benchmarks we clearly outperform the state-of-the-art. In recent years, the effect of the curse of high dimensionality has been studied in great detail on several problems such as clustering, nearest neighbor search, and indexing. In high dimensional space the data becomes sparse, and traditional indexing and algorithmic techniques fail from a efficiency and or effectiveness perspective. Recent research results show that in high dimensional space, the concept of proximity, distance or nearest neighbor may not even be qualitatively meaningful. In this paper, we view the dimensionality curse from the point of view of the distance metrics which are used to measure the similarity between objects. We specifically examine the behavior of the commonly used Lk norm and show that the problem of meaningfulness in high dimensionality is sensitive to the value of k. For example, this means that the Manhattan distance metric (L1 norm) is consistently more preferable than the Euclidean distance metric (L2 norm) for high dimensional data mining applications. Using the intuition derived from our analysis, we introduce and examine a natural extension of the Lk norm to fractional distance metrics. We show that the fractional distance metric provides more meaningful results both from the theoretical and empirical perspective. The results show that fractional distance metrics can significantly improve the effectiveness of standard clustering algorithms such as the k-means algorithm. We present a formulation of deep learning that aims at producing a large margin classifier. The notion of margin, minimum distance to a decision boundary, has served as the foundation of several theoretically profound and empirically successful results for both classification and regression tasks. However, most large margin algorithms are applicable only to shallow models with a preset feature representation; and conventional margin methods for neural networks only enforce margin at the output layer. Such methods are therefore not well suited for deep networks. In this work, we propose a novel loss function to impose a margin on any chosen set of layers of a deep network (including input and hidden layers). Our formulation allows choosing any norm on the metric measuring the margin. We demonstrate that the decision boundary obtained by our loss has nice properties compared to standard classification loss functions. Specifically, we show improved empirical results on the MNIST, CIFAR-10 and ImageNet datasets on multiple tasks: generalization from small training sets, corrupted labels, and robustness against adversarial perturbations. The resulting loss is general and complementary to existing data augmentation (such as random adversarial input transform) and regularization techniques (such as weight decay, dropout, and batch norm).