Metric (mathematics)

About: Metric (mathematics) is a research topic. Over the lifetime, 42617 publications have been published within this topic receiving 836571 citations. The topic is also known as: distance function & metric.

The geometry of proper scoring rules

Abstract: A decision problem is defined in terms of an outcome space, an action space and a loss function. Starting from these simple ingredients, we can construct: Proper Scoring Rule; Entropy Function; Divergence Function; Riemannian Metric; and Unbiased Estimating Equation. From an abstract viewpoint, the loss function defines a duality between the outcome and action spaces, while the correspondence between a distribution and its Bayes act induces a self-duality. Together these determine a “decision geometry” for the family of distributions on outcome space. This allows generalisation of many standard statistical concepts and properties. In particular we define and study generalised exponential families. Several examples are analysed, including a general Bregman geometry.

Attention-Based Ensemble for Deep Metric Learning

Abstract: Deep metric learning aims to learn an embedding function, modeled as deep neural network. This embedding function usually puts semantically similar images close while dissimilar images far from each other in the learned embedding space. Recently, ensemble has been applied to deep metric learning to yield state-of-the-art results. As one important aspect of ensemble, the learners should be diverse in their feature embeddings. To this end, we propose an attention-based ensemble, which uses multiple attention masks, so that each learner can attend to different parts of the object. We also propose a divergence loss, which encourages diversity among the learners. The proposed method is applied to the standard benchmarks of deep metric learning and experimental results show that it outperforms the state-of-the-art methods by a significant margin on image retrieval tasks.

Survey on distance metric learning and dimensionality reduction in data mining

Abstract: Distance metric learning is a fundamental problem in data mining and knowledge discovery. Many representative data mining algorithms, such as $$k$$ k -nearest neighbor classifier, hierarchical clustering and spectral clustering, heavily rely on the underlying distance metric for correctly measuring relations among input data. In recent years, many studies have demonstrated, either theoretically or empirically, that learning a good distance metric can greatly improve the performance of classification, clustering and retrieval tasks. In this survey, we overview existing distance metric learning approaches according to a common framework. Specifically, depending on the available supervision information during the distance metric learning process, we categorize each distance metric learning algorithm as supervised, unsupervised or semi-supervised. We compare those different types of metric learning methods, point out their strength and limitations. Finally, we summarize open challenges in distance metric learning and propose future directions for distance metric learning.

Image Set Classification Using Holistic Multiple Order Statistics Features and Localized Multi-kernel Metric Learning

Abstract: This paper presents a new approach for image set classification, where each training and testing example contains a set of image instances of an object captured from varying viewpoints or under varying illuminations. While a number of image set classification methods have been proposed in recent years, most of them model each image set as a single linear subspace or mixture of linear subspaces, which may lose some discriminative information for classification. To address this, we propose exploring multiple order statistics as features of image sets, and develop a localized multi-kernel metric learning (LMKML) algorithm to effectively combine different order statistics information for classification. Our method achieves the state-of-the-art performance on four widely used databases including the Honda/UCSD, CMU Mobo, and Youtube face datasets, and the ETH-80 object dataset.

Maximum Density Divergence for Domain Adaptation

Abstract: Unsupervised domain adaptation addresses the problem of transferring knowledge from a well-labeled source domain to an unlabeled target domain where the two domains have distinctive data distributions. Thus, the essence of domain adaptation is to mitigate the distribution divergence between the two domains. The state-of-the-art methods practice this very idea by either conducting adversarial training or minimizing a metric which defines the distribution gaps. In this paper, we propose a new domain adaptation method named adversarial tight match (ATM) which enjoys the benefits of both adversarial training and metric learning. Specifically, at first, we propose a novel distance loss, named maximum density divergence (MDD), to quantify the distribution divergence. MDD minimizes the inter-domain divergence (“match” in ATM) and maximizes the intra-class density (“tight” in ATM). Then, to address the equilibrium challenge issue in adversarial domain adaptation, we consider leveraging the proposed MDD into adversarial domain adaptation framework. At last, we tailor the proposed MDD as a practical learning loss and report our ATM. Both empirical evaluation and theoretical analysis are reported to verify the effectiveness of the proposed method. The experimental results on four benchmarks, both classical and large-scale, show that our method is able to achieve new state-of-the-art performance on most evaluations.