Recognition of objects with subtle differences has been used in many practical applications, such as car model recognition and maritime vessel identification. For discrimination of the objects in fine-grained detail, we focus on deep embedding learning by using a multi-task learning framework, in which the hierarchical labels (coarse and fine labels) of the samples are utilized both for classification and a quadruplet-based loss function. In order to improve the recognition strength of the learned features, we present a novel feature selection method specifically designed for four training samples of a quadruplet. By experiments, it is observed that the selection of very hard negative samples with relatively easy positive ones from the same coarse and fine classes significantly increases some performance metrics in a fine-grained dataset when compared to selecting the quadruplet samples randomly. The feature embedding learned by the proposed method achieves favorable performance against its state-of-the-art counterparts.
Earlier works on metric learning are based on @cite_19 . In that study, two identical neural networks extract the features of two arbitrary images. Next, these features are compared by a metric which is based on a radial function The distance between any two members in the feature space is defined as the cosine of the angle between them @cite_19 . . While their loss function forces the samples in the same class to be closer to each other in the sense of the selected distance function, the samples in the different classes are forced to be mapped far from each other. The cost function of such a network is given below @cite_2 where @math represents the operation of @math , and @math are distances in between samples.
@cite_19 @cite_2
2127589108 2138621090
We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Image-generating machine learning models are typically trained with loss functions based on distance in the image space. This often leads to over-smoothed results. We propose a class of loss functions, which we call deep perceptual similarity metrics (DeePSiM), that mitigate this problem. Instead of computing distances in the image space, we compute distances between image features extracted by deep neural networks. This metric better reflects perceptually similarity of images and thus leads to better results. We show three applications: autoencoder training, a modification of a variational autoencoder, and inversion of deep convolutional networks. In all cases, the generated images look sharp and resemble natural images. Let @math be a set of elements and d a distance function defined among them. Let NNk(u) be the k elements in @math having the smallest distance to u. The k-nearest neighbor graph (knng) is a weighted directed graph @math such that E= (u,v), v∈NNk(u) . Several knng construction algorithms are known, but they are not suitable to general metric spaces. We present a general methodology to construct knngs that exploits several features of metric spaces. Experiments suggest that it yields costs of the form c1n1.27 distance computations for low and medium dimensional spaces, and c2n1.90 for high dimensional ones. We propose a class of loss functions, which we call deep perceptual similarity metrics (DeePSiM), allowing to generate sharp high resolution images from compressed abstract representations. Instead of computing distances in the image space, we compute distances between image features extracted by deep neural networks. This metric reflects perceptual similarity of images much better and, thus, leads to better results. We demonstrate two examples of use cases of the proposed loss: (1) networks that invert the AlexNet convolutional network; (2) a modified version of a variational autoencoder that generates realistic high-resolution random images. Abstract Irregular sampling and “stable sampling” of band-limited functions have been studied by H.J. Landau [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52]. We prove that quasicrystals are sets of stable sampling. To cite this article: B. Matei, Y. Meyer, C. R. Acad. Sci. Paris, Ser. I 346 (2008). Let @math denote a set of polynomials with complex coefficients. Let @math denote any set of sample points . For any @math , the discrete polynomial transform of @math (with respect to @math and @math ) is defined as the collection of sums, @math , where @math for some associated weight function @math . These sorts of transforms find important applications in areas such as medical imaging and signal processing. In this paper, we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of @math orthogonal polynomials of degree at most @math , we give an @math algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the @math operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a three-term recurrence and thus it may be applied to any such set of discretely sampled functions. In particular, sampled orthogonal polynomials generate the vector space of functions on a distance transitive graph. As a direct application of our work, we are able to give a fast algorithm for computing subspace decompositions of this vector space which respect the action of the symmetry group of such a graph. This has direct applications to treating computational bottlenecks in the spectral analysis of data on distance transitive graphs, and we discuss this in some detail. Person re-identification (re-id) aims to match pedestrians observed by disjoint camera views. It attracts increasing attention in computer vision due to its importance to surveillance systems. To combat the major challenge of cross-view visual variations, deep embedding approaches are proposed by learning a compact feature space from images such that the Euclidean distances correspond to their cross-view similarity metric. However, the global Euclidean distance cannot faithfully characterize the ideal similarity in a complex visual feature space because features of pedestrian images exhibit unknown distributions due to large variations in poses, illumination and occlusion. Moreover, intra-personal training samples within a local range which are robust to guide deep embedding against uncontrolled variations cannot be captured by a global Euclidean distance. In this paper, we study the problem of person re-id by proposing a novel sampling to mine suitable positives (i.e., intra-class) within a local range to improve the deep embedding in the context of large intra-class variations. Our method is capable of learning a deep similarity metric adaptive to local sample structure by minimizing each sample's local distances while propagating through the relationship between samples to attain the whole intra-class minimization. To this end, a novel objective function is proposed to jointly optimize similarity metric learning, local positive mining and robust deep feature embedding. This attains local discriminations by selecting local-ranged positive samples, and the learned features are robust to dramatic intra-class variations. Experiments on benchmarks show state-of-the-art results achieved by our method. (C) 2017 Elsevier Ltd. All rights reserved. Person recognition aims at recognizing the same identity across time and space with complicated scenes and similar appearance. In this paper, we propose a novel method to address this task by training a network to obtain robust and representative features. The intuition is that we directly compare and optimize the cosine distance between two features - enlarging inter-class distinction as well as alleviating inner-class variance. We propose a congenerous cosine loss by minimizing the cosine distance between samples and their cluster centroid in a cooperative way. Such a design reduces the complexity and could be implemented via softmax with normalized inputs. Our method also differs from previous work in person recognition that we do not conduct a second training on the test subset. The identity of a person is determined by measuring the similarity from several body regions in the reference set. Experimental results show that the proposed approach achieves better classification accuracy against previous state-of-the-arts. Metric learning aims to construct an embedding where two extracted features corresponding to the same identity are likely to be closer than features from different identities. This paper presents a method for learning such a feature space where the cosine similarity is effectively optimized through a simple re-parametrization of the conventional softmax classification regime. At test time, the final classification layer can be stripped from the network to facilitate nearest neighbor queries on unseen individuals using the cosine similarity metric. This approach presents a simple alternative to direct metric learning objectives such as siamese networks that have required sophisticated pair or triplet sampling strategies in the past. The method is evaluated on two large-scale pedestrian re-identification datasets where competitive results are achieved overall. In particular, we achieve better generalization on the test set compared to a network trained with triplet loss. A standard result in metric geometry is that every compact geodesic metric space can be approximated arbitrarily well by finite metric graphs in the Gromov-Hausdorff sense. It is well known that the first Betti number of the approximating graphs may blow up as the approximation gets finer. In our work, given a compact geodesic metric space @math , we define a sequence @math of non-negative real numbers by @math By construction, and the above result, this is a non-increasing sequence with limit @math . We study this sequence and its rates of decay with @math . We also identify a precise relationship between the sequence and the first Vietoris-Rips persistence barcode of @math . Furthermore, we specifically analyze @math and find upper and lower bounds based on hyperbolicity and other metric invariants. As a consequence of the tools we develop, our work also provides a Gromov-Hausdorff stability result for the Reeb construction on geodesic metric spaces with respect to the function given by distance to a reference point. Person re-identification is challenging due to the large variations of pose, illumination, occlusion and camera view. Owing to these variations, the pedestrian data is distributed as highly-curved manifolds in the feature space, despite the current convolutional neural networks (CNN)’s capability of feature extraction. However, the distribution is unknown, so it is difficult to use the geodesic distance when comparing two samples. In practice, the current deep embedding methods use the Euclidean distance for the training and test. On the other hand, the manifold learning methods suggest to use the Euclidean distance in the local range, combining with the graphical relationship between samples, for approximating the geodesic distance. From this point of view, selecting suitable positive (i.e. intra-class) training samples within a local range is critical for training the CNN embedding, especially when the data has large intra-class variations. In this paper, we propose a novel moderate positive sample mining method to train robust CNN for person re-identification, dealing with the problem of large variation. In addition, we improve the learning by a metric weight constraint, so that the learned metric has a better generalization ability. Experiments show that these two strategies are effective in learning robust deep metrics for person re-identification, and accordingly our deep model significantly outperforms the state-of-the-art methods on several benchmarks of person re-identification. Therefore, the study presented in this paper may be useful in inspiring new designs of deep models for person re-identification. The interleaving distance was originally defined in the field of Topological Data Analysis (TDA) by as a metric on the class of persistence modules parametrized over the real line. subsequently extended the definition to categories of functors on a poset, the objects in these categories being regarded as generalized persistence modules'. These metrics typically depend on the choice of a lax semigroup of endomorphisms of the poset. The purpose of the present paper is to develop a more general framework for the notion of interleaving distance using the theory of actegories'. Specifically, we extend the notion of interleaving distance to arbitrary categories equipped with a flow, i.e. a lax monoidal action by the monoid @math . In this way, the class of objects in such a category acquires the structure of a Lawvere metric space. Functors that are colax @math -equivariant yield maps that are @math -Lipschitz. This leads to concise proofs of various known stability results from TDA, by considering appropriate colax @math -equivariant functors. Along the way, we show that several common metrics, including the Hausdorff distance and the @math -norm, can be realized as interleaving distances in this general perspective. A metric space X is said to be absolutely Lipschitz extendable if every Lipschitz function f from X into any Banach space Z can be extended to any containing space Y⊇X, where the loss in the Lipschitz constant in the extension is independent of Y,Z, and f. We show that various classes of natural metric spaces are absolutely Lipschitz extendable. To cite this article: J.R. Lee, A. Naor, C. R. Acad. Sci. Paris, Ser. I 338 (2004). Most existing 3D object recognition algorithms focus on leveraging the strong discriminative power of deep learning models with softmax loss for the classification of 3D data, while learning discriminative features with deep metric learning for 3D object retrieval is more or less neglected. In the paper, we study variants of deep metric learning losses for 3D object retrieval, which did not receive enough attention from this area. First , two kinds of representative losses, triplet loss and center loss, are introduced which could learn more discriminative features than traditional classification loss. Then, we propose a novel loss named triplet-center loss, which can further enhance the discriminative power of the features. The proposed triplet-center loss learns a center for each class and requires that the distances between samples and centers from the same class are closer than those from different classes. Extensive experimental results on two popular 3D object retrieval benchmarks and two widely-adopted sketch-based 3D shape retrieval benchmarks consistently demonstrate the effectiveness of our proposed loss, and significant improvements have been achieved compared with the state-of-the-arts. Most existing 3D object recognition algorithms focus on leveraging the strong discriminative power of deep learning models with softmax loss for the classification of 3D data, while learning discriminative features with deep metric learning for 3D object retrieval is more or less neglected. In the paper, we study variants of deep metric learning losses for 3D object retrieval, which did not receive enough attention from this area. First, two kinds of representative losses, triplet loss and center loss, are introduced which could learn more discriminative features than traditional classification loss. Then, we propose a novel loss named triplet-center loss, which can further enhance the discriminative power of the features. The proposed triplet-center loss learns a center for each class and requires that the distances between samples and centers from the same class are closer than those from different classes. Extensive experimental results on two popular 3D object retrieval benchmarks and two widely-adopted sketch-based 3D shape retrieval benchmarks consistently demonstrate the effectiveness of our proposed loss, and significant improvements have been achieved compared with the state-of-the-arts. Learning the distance metric between pairs of examples is of great importance for learning and visual recognition. With the remarkable success from the state of the art convolutional neural networks, recent works [1, 31] have shown promising results on discriminatively training the networks to learn semantic feature embeddings where similar examples are mapped close to each other and dissimilar examples are mapped farther apart. In this paper, we describe an algorithm for taking full advantage of the training batches in the neural network training by lifting the vector of pairwise distances within the batch to the matrix of pairwise distances. This step enables the algorithm to learn the state of the art feature embedding by optimizing a novel structured prediction objective on the lifted problem. Additionally, we collected Stanford Online Products dataset: 120k images of 23k classes of online products for metric learning. Our experiments on the CUB-200-2011 [37], CARS196 [19], and Stanford Online Products datasets demonstrate significant improvement over existing deep feature embedding methods on all experimented embedding sizes with the GoogLeNet [33] network. The source code and the dataset are available at: https: github.com rksltnl Deep-Metric-Learning-CVPR16. This paper addresses deep face recognition (FR) problem under open-set protocol, where ideal face features are expected to have smaller maximal intra-class distance than minimal inter-class distance under a suitably chosen metric space. However, few existing algorithms can effectively achieve this criterion. To this end, we propose the angular softmax (A-Softmax) loss that enables convolutional neural networks (CNNs) to learn angularly discriminative features. Geometrically, A-Softmax loss can be viewed as imposing discriminative constraints on a hypersphere manifold, which intrinsically matches the prior that faces also lie on a manifold. Moreover, the size of angular margin can be quantitatively adjusted by a parameter @math . We further derive specific @math to approximate the ideal feature criterion. Extensive analysis and experiments on Labeled Face in the Wild (LFW), Youtube Faces (YTF) and MegaFace Challenge show the superiority of A-Softmax loss in FR tasks. The code has also been made publicly available. Convolutional neural networks have achieved great improvement on face recognition in recent years because of its extraordinary ability in learning discriminative features of people with different identities. To train such a well-designed deep network, tremendous amounts of data is indispensable. Long tail distribution specifically refers to the fact that a small number of generic entities appear frequently while other objects far less existing. Considering the existence of long tail distribution of the real world data, large but uniform distributed data are usually hard to retrieve. Empirical experiences and analysis show that classes with more samples will pose greater impact on the feature learning process and inversely cripple the whole models feature extracting ability on tail part data. Contrary to most of the existing works that alleviate this problem by simply cutting the tailed data for uniform distributions across the classes, this paper proposes a new loss function called range loss to effectively utilize the whole long tailed data in training process. More specifically, range loss is designed to reduce overall intra-personal variations while enlarging inter-personal differences within one mini-batch simultaneously when facing even extremely unbalanced data. The optimization objective of range loss is the @math greatest range's harmonic mean values in one class and the shortest inter-class distance within one batch. Extensive experiments on two famous and challenging face recognition benchmarks (Labeled Faces in the Wild (LFW) and YouTube Faces (YTF) not only demonstrate the effectiveness of the proposed approach in overcoming the long tail effect but also show the good generalization ability of the proposed approach. We consider the problem of learning an unknown function @math on the @math -dimensional sphere with respect to the square loss, given i.i.d. samples @math where @math is a feature vector uniformly distributed on the sphere and @math . We study two popular classes of models that can be regarded as linearizations of two-layers neural networks around a random initialization: (RF) The random feature model of Rahimi-Recht; (NT) The neural tangent kernel model of Jacot-Gabriel-Hongler. Both these approaches can also be regarded as randomized approximations of kernel ridge regression (with respect to different kernels), and hence enjoy universal approximation properties when the number of neurons @math diverges, for a fixed dimension @math . We prove that, if both @math and @math are large, the behavior of these models is instead remarkably simpler. If @math , then RF performs no better than linear regression with respect to the raw features @math , and NT performs no better than linear regression with respect to degree-one and two monomials in the @math . More generally, if @math then RF fits at most a degree- @math polynomial in the raw features, and NT fits at most a degree- @math polynomial. This paper addresses the basic question of how well a tree can approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of @math , with the guarantee that for each @math simultaneously, the distortion of a fraction @math of all pairs is bounded accordingly. Quantitatively, we prove that any finite metric space embeds into an ultrametric with scaling distortion @math . For the graph setting, we prove that any weighted graph contains a spanning tree with scaling distortion @math . These bounds are tight even for embedding into arbitrary trees. These results imply that the average distortion of the embedding is constant and that the @math distortion is @math . For probabilistic embedding into spanning trees we prove a scaling distortion of @math , which implies constant @math -distortion for every fixed @math .