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.

Hamming Distance Metric Learning

Abstract: Motivated by large-scale multimedia applications we propose to learn mappings from high-dimensional data to binary codes that preserve semantic similarity. Binary codes are well suited to large-scale applications as they are storage efficient and permit exact sub-linear kNN search. The framework is applicable to broad families of mappings, and uses a flexible form of triplet ranking loss. We overcome discontinuous optimization of the discrete mappings by minimizing a piecewise-smooth upper bound on empirical loss, inspired by latent structural SVMs. We develop a new loss-augmented inference algorithm that is quadratic in the code length. We show strong retrieval performance on CIFAR-10 and MNIST, with promising classification results using no more than kNN on the binary codes.

The Space of Kähler Metrics

Abstract: This paper, the second of a series, deals with the function space \mathcal{H} of all smooth Kahler metrics in any given n-dimensional, closed complex manifold V, these metrics being restricted to a given, fixed, real cohomology class, called a polarization of V. This function space is equipped with a pre-Hilbert metric structure introduced by T. Mabuchi [10], who also showed that, formally, this metric has nonpositive curvature. In the first paper of this series [4], the second author showed that the same space is a path length space. He also proved that \mathcal{H} is geodesically convex in the sense that, for any two points of \mathcal{H}, there is a unique geodesic path joining them, which is always length minimizing and of class C1,1. This partially verifies two conjectures of Donaldson [8] on the subject. In the present paper, we show first of all, that the space is, as expected, a path length space of nonpositive curvature in the sense of A. D. Aleksandrov. A second result is related to the theory of extremal Kahler metrics, namely that the gradient flow in \mathcal{H} of the "K energy" of V has the property that it strictly decreases the length of all paths in \mathcal{H}, except those induced by one parameter families of holomorphic automorphisms of M.

Learning a Discriminative Null Space for Person Re-identification

Abstract: Most existing person re-identification (re-id) methods focus on learning the optimal distance metrics across camera views. Typically a person's appearance is represented using features of thousands of dimensions, whilst only hundreds of training samples are available due to the difficulties in collecting matched training images. With the number of training samples much smaller than the feature dimension, the existing methods thus face the classic small sample size (SSS) problem and have to resort to dimensionality reduction techniques and/or matrix regularisation, which lead to loss of discriminative power. In this work, we propose to overcome the SSS problem in re-id distance metric learning by matching people in a discriminative null space of the training data. In this null space, images of the same person are collapsed into a single point thus minimising the within-class scatter to the extreme and maximising the relative between-class separation simultaneously. Importantly, it has a fixed dimension, a closed-form solution and is very efficient to compute. Extensive experiments carried out on five person re-identification benchmarks including VIPeR, PRID2011, CUHK01, CUHK03 and Market1501 show that such a simple approach beats the state-of-the-art alternatives, often by a big margin.