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.

Distance metric learning with eigenvalue optimization

Abstract: 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 (Weinberger et al., 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).

Automated Derivation of Primitives for Movement Classification

Abstract: We describe a new method for representing human movement compactly, in terms of a linear super-imposition of simpler movements termed i>primitives. This method is a part of a larger research project aimed at modeling motor control and imitation using the notion of perceptuo-motor primitives, a basis set of coupled perceptual and motor routines. In our model, the perceptual system is biased by the set of motor behaviors the agent can execute. Thus, an agent can automatically classify observed movements into its executable repertoire. In this paper, we describe a method for automatically deriving a set of primitives directly from human movement data. We used movement data gathered from a psychophysical experiment on human imitation to derive the primitives. The data were first filtered, then segmented, and principal component analysis was applied to the segments. The eigenvectors corresponding to a few of the highest eigenvalues provide us with a basis set of primitives. These are used, through superposition and sequencing, to reconstruct the training movements as well as novel ones. The validation of the method was performed on a humanoid simulation with physical dynamics. The effectiveness of the motion reconstruction was measured through an error metric. We also explored and evaluated a technique of clustering in the space of primitives for generating controllers for executing frequently used movements.

Generalized geometry and M theory

Abstract: We reformulate the Hamiltonian form of bosonic eleven dimensional supergravity in terms of an object that unifies the three-form and the metric. For the case of four spatial dimensions, the duality group is manifest and the metric and C-field are on an equal footing even though no dimensional reduction is required for our results to hold. One may also describe our results using the generalized geometry that emerges from membrane duality. The relationship between the twisted Courant algebra and the gauge symmetries of eleven dimensional supergravity are described in detail.

Distance metrics and band selection in hyperspectral processing with applications to material identification and spectral libraries

Abstract: At the core of most hyperspectral processing algorithms are distance metrics that compare two spectra and return a scalar value based on some notion of similarity. The two most common distance metrics in hyperspectral processing are the spectral angle mapper (SAM) and the Euclidean minimum distance (EMD), and each metric possesses distinct mathematical and physical properties. In this paper, we enumerate the characteristics of both metrics, and, based on an exact decomposition of SAM, we derive a technique called band add-on (BAO) that iteratively selects bands to increase the angular separation between two spectra. Unlike other feature selection algorithms, BAO exploits a mathematical decomposition of SAM to incrementally add bands. We extend BAO to the more practical problem of increasing the angular separability between two classes of spectra. This scenario parallels the material identification problem where quite often only a small number (<10) of ground-truth measurements are collected for each material class, and statistical classification methods are inapplicable. Two algorithms for selecting bands and class templates are presented to increase the angular separation between two classes. The techniques are compared with several other metric-based approaches in binary discrimination tests with real data. The results demonstrate that band selection can improve the discrimination of very similar targets, while using only a fraction of the available spectral bands.