For many tasks and data types, there are natural transformations to which the data should be invariant or insensitive. For instance, in visual recognition, natural images should be insensitive to rotation and translation. This requirement and its implications have been important in many machine learning applications, and tolerance for image transformations was primarily achieved by using robust feature vectors. In this paper we propose a novel and computationally efficient way to learn a local Mahalanobis metric per datum, and show how we can learn a local invariant metric to any transformation in order to improve performance.
Metric learning is an active research field with many algorithms, generally divided into linear @cite_20 which learn a Mahalanobis distance, non-linear @cite_12 that learn a nonlinear transformation and use @math distance on the transformed space, and local which learn a metric per datum. The LMNN and MLMM @cite_20 algorithm are considered the leading metric learning method. For a recent comprehensive survey that covers linear, non-linear and local methods see @cite_19 .
@cite_19 @cite_12 @cite_20
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Metric learning has become a very active research field. The most popular representative–Mahalanobis metric learning–can be seen as learning a linear transformation and then computing the Euclidean metric in the transformed space. Since a linear transformation might not always be appropriate for a given learning problem, kernelized versions of various metric learning algorithms exist. However, the problem then becomes finding the appropriate kernel function. Multiple kernel learning addresses this limitation by learning a linear combination of a number of predefined kernels; this approach can be also readily used in the context of multiple-source learning to fuse different data sources. Surprisingly, and despite the extensive work on multiple kernel learning for SVMs, there has been no work in the area of metric learning with multiple kernel learning. In this paper we fill this gap and present a general approach for metric learning with multiple kernel learning. Our approach can be instantiated with different metric learning algorithms provided that they satisfy some constraints. Experimental evidence suggests that our approach outperforms metric learning with an unweighted kernel combination and metric learning with cross-validation based kernel selection. Metric learning aims at finding a distance that approximates a task-specific notion of semantic similarity. Typically, a Mahalanobis distance is learned from pairs of data labeled as being semantically similar or not. In this paper, we learn such metrics in a weakly supervised setting where "bags" of instances are labeled with "bags" of labels. We formulate the problem as a multiple instance learning (MIL) problem over pairs of bags. If two bags share at least one label, we label the pair positive, and negative otherwise. We propose to learn a metric using those labeled pairs of bags, leading to MildML, for multiple instance logistic discriminant metric learning. MildML iterates between updates of the metric and selection of putative positive pairs of examples from positive pairs of bags. To evaluate our approach, we introduce a large and challenging data set, Labeled Yahoo! News, which we have manually annotated and contains 31147 detected faces of 5873 different people in 20071 images. We group the faces detected in an image into a bag, and group the names detected in the caption into a corresponding set of labels. When the labels come from manual annotation, we find that MildML using the bag-level annotation performs as well as fully supervised metric learning using instance-level annotation. We also consider performance in the case of automatically extracted labels for the bags, where some of the bag labels do not correspond to any example in the bag. In this case MildML works substantially better than relying on noisy instance-level annotations derived from the bag-level annotation by resolving face-name associations in images with their captions. In this paper, we introduce two novel metric learning algorithms, Χ2-LMNN and GB-LMNN, which are explicitly designed to be non-linear and easy-to-use. The two approaches achieve this goal in fundamentally different ways: Χ2-LMNN inherits the computational benefits of a linear mapping from linear metric learning, but uses a non-linear Χ2-distance to explicitly capture similarities within histogram data sets; GB-LMNN applies gradient-boosting to learn non-linear mappings directly in function space and takes advantage of this approach's robustness, speed, parallelizability and insensitivity towards the single additional hyper-parameter. On various benchmark data sets, we demonstrate these methods not only match the current state-of-the-art in terms of kNN classification error, but in the case of Χ2-LMNN, obtain best results in 19 out of 20 learning settings. 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 Metric learning makes it plausible to learn semantically meaningful distances for complex distributions of data using label or pairwise constraint information. However, to date, most metric learning methods are based on a single Mahalanobis metric, which cannot handle heterogeneous data well. Those that learn multiple metrics throughout the feature space have demonstrated superior accuracy, but at a severe cost to computational efficiency. Here, we adopt a new angle on the metric learning problem and learn a single metric that is able to implicitly adapt its distance function throughout the feature space. This metric adaptation is accomplished by using a random forest-based classifier to underpin the distance function and incorporate both absolute pairwise position and standard relative position into the representation. We have implemented and tested our method against state of the art global and multi-metric methods on a variety of data sets. Overall, the proposed method outperforms both types of method in terms of accuracy (consistently ranked first) and is an order of magnitude faster than state of the art multi-metric methods (16x faster in the worst case). 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 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 and kernel learning arise in several machine learning applications. However, most existing metric learning algorithms are limited to learning metrics over low-dimensional data, while existing kernel learning algorithms are often limited to the transductive setting and do not generalize to new data points. In this paper, we study the connections between metric learning and kernel learning that arise when studying metric learning as a linear transformation learning problem. In particular, we propose a general optimization framework for learning metrics via linear transformations, and analyze in detail a special case of our framework-that of minimizing the LogDet divergence subject to linear constraints. We then propose a general regularized framework for learning a kernel matrix, and show it to be equivalent to our metric learning framework. Our theoretical connections between metric and kernel learning have two main consequences: 1) the learned kernel matrix parameterizes a linear transformation kernel function and can be applied inductively to new data points, 2) our result yields a constructive method for kernelizing most existing Mahalanobis metric learning formulations. We demonstrate our learning approach by applying it to large-scale real world problems in computer vision, text mining and semi-supervised kernel dimensionality reduction. 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. Linear metric learning is a widely used methodology to learn a dissimilarity function from a set of similar dissimilar example pairs. Using a single metric may be a too restrictive assumption when handling heterogeneous datasets. Recently, local metric learning methods have been introduced to overcome this limitation. However, they are subjects to constraints preventing their usage in many applications. For example, they require knowledge of the class label of the training points. In this paper, we present a novel local metric learning method, which overcomes some limitations of previous approaches. The method first computes a Gaussian Mixture Model from a low dimensional embedding of training data. Then it estimates a set of local metrics by solving a convex optimization problem; finally, a dissimilarity function is obtained by aggregating the local metrics. Our experiments show that the proposed method achieves state-of-the-art results on four datasets. We propose a new approach for metric learning by framing it as learning a sparse combination of locally discriminative metrics that are inexpensive to generate from the training data. This flexible framework allows us to naturally derive formulations for global, multi-task and local metric learning. The resulting algorithms have several advantages over existing methods in the literature: a much smaller number of parameters to be estimated and a principled way to generalize learned metrics to new testing data points. To analyze the approach theoretically, we derive a generalization bound that justifies the sparse combination. Empirically, we evaluate our algorithms on several datasets against state-of-the-art metric learning methods. The results are consistent with our theoretical findings and demonstrate the superiority of our approach in terms of classification performance and scalability. This paper introduces a novel classification approach which improves the performance of support vector machines (SVMs) by learning a distance metric. The metric learned is a Mahalanobis metric previously trained so that examples from different classes are separated with a large margin. The learned metric is used to define a kernel function for SVM classification. In this context, the metric can be seen as a linear transformation of the original inputs before applying an SVM classifier that uses Euclidean distances. This transformation increases the separability of classes in the transformed space where the classification is applied. Experiments demonstrate significant improvements in classification tasks on various data sets. We study metric learning as a problem of information retrieval. We present a general metric learning algorithm, based on the structural SVM framework, to learn a metric such that rankings of data induced by distance from a query can be optimized against various ranking measures, such as AUC, Precision-at-k, MRR, MAP or NDCG. We demonstrate experimental results on standard classification data sets, and a large-scale online dating recommendation problem. The main theme of this paper is to develop a novel eigenvalue optimization framework for learning a Mahalanobis metric. Within this context, we introduce a novel metric learning approach called DML-eig which is shown to be equivalent to a well-known eigenvalue optimization problem called minimizing the maximal eigenvalue of a symmetric matrix (Overton, 1988; Lewis and Overton, 1996). Moreover, we formulate LMNN (, 2005), one of the state-of-the-art metric learning methods, as a similar eigenvalue optimization problem. This novel framework not only provides new insights into metric learning but also opens new avenues to the design of efficient metric learning algorithms. Indeed, first-order algorithms are developed for DML-eig and LMNN which only need the computation of the largest eigenvector of a matrix per iteration. Their convergence characteristics are rigorously established. Various experiments on benchmark data sets show the competitive performance of our new approaches. In addition, we report an encouraging result on a difficult and challenging face verification data set called Labeled Faces in the Wild (LFW). 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. 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. 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. Metric learning algorithms can provide useful distance functions for a variety of domains, and recent work has shown good accuracy for problems where the learner can access all distance constraints at once. However, in many real applications, constraints are only available incrementally, thus necessitating methods that can perform online updates to the learned metric. Existing online algorithms offer bounds on worst-case performance, but typically do not perform well in practice as compared to their offline counterparts. We present a new online metric learning algorithm that updates a learned Mahalanobis metric based on LogDet regularization and gradient descent. We prove theoretical worst-case performance bounds, and empirically compare the proposed method against existing online metric learning algorithms. To further boost the practicality of our approach, we develop an online locality-sensitive hashing scheme which leads to efficient updates to data structures used for fast approximate similarity search. We demonstrate our algorithm on multiple datasets and show that it outperforms relevant baselines. This paper presents a new discriminative deep metric learning (DDML) method for face verification in the wild. Different from existing metric learning-based face verification methods which aim to learn a Mahalanobis distance metric to maximize the inter-class variations and minimize the intra-class variations, simultaneously, the proposed DDML trains a deep neural network which learns a set of hierarchical nonlinear transformations to project face pairs into the same feature subspace, under which the distance of each positive face pair is less than a smaller threshold and that of each negative pair is higher than a larger threshold, respectively, so that discriminative information can be exploited in the deep network. Our method achieves very competitive face verification performance on the widely used LFW and YouTube Faces (YTF) datasets. Multi-metric learning techniques learn local metric tensors in different parts of a feature space. With such an approach, even simple classifiers can be competitive with the state-of-the-art because the distance measure locally adapts to the structure of the data. The learned distance measure is, however, non-metric, which has prevented multi-metric learning from generalizing to tasks such as dimensionality reduction and regression in a principled way. We prove that, with appropriate changes, multi-metric learning corresponds to learning the structure of a Riemannian manifold. We then show that this structure gives us a principled way to perform dimensionality reduction and regression according to the learned metrics. Algorithmically, we provide the first practical algorithm for computing geodesics according to the learned metrics, as well as algorithms for computing exponential and logarithmic maps on the Riemannian manifold. Together, these tools let many Euclidean algorithms take advantage of multi-metric learning. We illustrate the approach on regression and dimensionality reduction tasks that involve predicting measurements of the human body from shape data.