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 convergence of variable metric methods for nonlinearly constrained optimization calculations

Abstract: Variable metric methods for unconstrained optimization calculations can be extended to the constrained case by regarding the positive definite matrix that is revised on each iteration as an approximation to the second derivative matrix of the Lagrangian function. Linear approximations to the constraints are used. Han (1976) has analyzed the convergence of these methods in the case when the true second derivative matrix of the Lagrangian function is positive definite at the solution. However, this matrix sometimes has negative eigenvalues so we analyze the rate of convergence in this case. We find that it is still superlinear. Therefore we may continue to use positive definite second derivative approximations and there is no need to introduce any penalty terms. The given theory helps to explain the excellent numerical results that are obtained by a recent algorithm (Powell, 1977).

Minimal Mutation Trees of Sequences

Abstract: Given a finite tree, some of whose vertices are identified with given finite sequences, we show how to construct sequences for all the remaining vertices simultaneously, so as to minimize the total edge-length of the tree. Edge-length is calculated by a metric whose biological significance is the mutational distance between two sequences.

Learning Locally-Adaptive Decision Functions for Person Verification

Abstract: This paper considers the person verification problem in modern surveillance and video retrieval systems. The problem is to identify whether a pair of face or human body images is about the same person, even if the person is not seen before. Traditional methods usually look for a distance (or similarity) measure between images (e.g., by metric learning algorithms), and make decisions based on a fixed threshold. We show that this is nevertheless insufficient and sub-optimal for the verification problem. This paper proposes to learn a decision function for verification that can be viewed as a joint model of a distance metric and a locally adaptive thresholding rule. We further formulate the inference on our decision function as a second-order large-margin regularization problem, and provide an efficient algorithm in its dual from. We evaluate our algorithm on both human body verification and face verification problems. Our method outperforms not only the classical metric learning algorithm including LMNN and ITML, but also the state-of-the-art in the computer vision community.

Fast Computation of Wasserstein Barycenters

Abstract: We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasserstein barycenters that build upon the subgradient method. A direct implementation of these algorithms is, however, too costly because it would require the repeated resolution of large primal and dual optimal transport problems to compute subgradients. Extending the work of Cuturi (2013), we propose to smooth the Wasserstein distance used in the definition of Wasserstein barycenters with an entropic regularizer and recover in doing so a strictly convex objective whose gradients can be computed for a considerably cheaper computational cost using matrix scaling algorithms. We use these algorithms to visualize a large family of images and to solve a constrained clustering problem.