Metric learning methods for dimensionality reduction in combination with k-Nearest Neighbors (kNN) have been extensively deployed in many classification, data embedding, and information retrieval applications. However, most of these approaches involve pairwise training data comparisons, and thus have quadratic computational complexity with respect to the size of training set, preventing them from scaling to fairly big datasets. Moreover, during testing, comparing test data against all the training data points is also expensive in terms of both computational cost and resources required. Furthermore, previous metrics are either too constrained or too expressive to be well learned. To effectively solve these issues, we present an exemplar-centered supervised shallow parametric data embedding model, using a Maximally Collapsing Metric Learning (MCML) objective. Our strategy learns a shallow high-order parametric embedding function and compares training test data only with learned or precomputed exemplars, resulting in a cost function with linear computational complexity for both training and testing. We also empirically demonstrate, using several benchmark datasets, that for classification in two-dimensional embedding space, our approach not only gains speedup of kNN by hundreds of times, but also outperforms state-of-the-art supervised embedding approaches.
 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. Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Component Analysis (PCA) – a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold. As a result, LPP shares many of the data representation properties of nonlinear techniques such as Laplacian Eigenmaps or Locally Linear Embedding. Yet LPP is linear and more crucially is defined everywhere in ambient space rather than just on the training data points. This is borne out by illustrative examples on some high dimensional data sets.   Deep learning has been successfully applied to perform non-linear embedding. In this paper, we present supervised embedding techniques that use a deep network to collapse classes. The network is pre-trained using a stack of RBMs, and finetuned using approaches that try to collapse classes. The finetuning is inspired by ideas from NCA, but it uses a Student t-distribution to model the similarities of data points belonging to the same class in the embedding. We investigate two types of objective functions: deep t-distributed MCML (dt-MCML) and deep t-distributed NCA (dt-NCA). Our experiments on two handwritten digit data sets reveal the strong performance of dt-MCML in supervised parametric data visualization, whereas dt-NCA outperforms alternative techniques when embeddings with more than two or three dimensions are constructed, e.g., to obtain good classification performances. Overall, our results demonstrate the advantage of using a deep architecture and a heavy-tailed t-distribution for measuring pairwise similarities in supervised embedding. We present a new technique called “t-SNE” that visualizes high-dimensional data by giving each datapoint a location in a two or three-dimensional map. The technique is a variation of Stochastic Neighbor Embedding (Hinton and Roweis, 2002) that is much easier to optimize, and produces significantly better visualizations by reducing the tendency to crowd points together in the center of the map. t-SNE is better than existing techniques at creating a single map that reveals structure at many different scales. This is particularly important for high-dimensional data that lie on several different, but related, low-dimensional manifolds, such as images of objects from multiple classes seen from multiple viewpoints. For visualizing the structure of very large datasets, we show how t-SNE can use random walks on neighborhood graphs to allow the implicit structure of all of the data to influence the way in which a subset of the data is displayed. We illustrate the performance of t-SNE on a wide variety of datasets and compare it with many other non-parametric visualization techniques, including Sammon mapping, Isomap, and Locally Linear Embedding. The visualizations produced by t-SNE are significantly better than those produced by the other techniques on almost all of the datasets.  The paper presents a new unsupervised dimensionality reduction technique, called parametric t-SNE, that learns a parametric mapping between the high-dimensional data space and the low-dimensional latent space. Parametric t-SNE learns the parametric mapping in such a way that the local structure of the data is preserved as well as possible in the latent space. We evaluate the performance of parametric t-SNE in experiments on three datasets, in which we compare it to the performance of two other unsupervised parametric dimensionality reduction techniques. The results of experiments illustrate the strong performance of parametric t-SNE, in particular, in learning settings in which the dimensionality of the latent space is relatively low. 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  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. 