Various problems in machine learning, databases, and statistics involve pairwise distances among a set of objects. It is often desirable for these distances to satisfy the properties of a metric, especially the triangle in- equality. Applications where metric data is useful include clustering, classification, metric-based indexing, and approximation algorithms for various graph problems. This paper presents the Metric Nearness Prob- lem: Given a dissimilarity matrix, find the "nearest" matrix of distances that satisfy the triangle inequalities. For p nearness measures, this pa- per develops efficient triangle fixing algorithms that compute globally optimal solutions by exploiting the inherent structure of the problem.

Classical multidimensional scaling (CMDS) is a technique that aims to embed a set of objects in a Euclidean space given their pairwise Euclidean distance matrix. The main part of CMDS is based on double centering a squared distance matrix and employing a truncated eigendecomposition to recover the point coordinates. A central result in CMDS connects the squared Euclidean matrix to a Gram matrix derived from the set of points. In this paper, we study a dual basis approach to classical multidimensional scaling. We give an explicit formula for the dual basis and fully characterize the spectrum of an essential matrix in the dual basis framework. We make connections to a related problem in metric nearness.

Given a dissimilarity matrix, the metric nearness problem is to find the nearest matrix of distances that satisfy the triangle inequalities. This problem has wide applications, such as sensor networks, image processing, and so on. But it is of great challenge even to obtain a moderately accurate solution due to the $O(n^{3})$ metric constraints and the nonsmooth objective function which is usually a weighted $\ell_{p}$ norm based distance. In this paper, we propose a delayed constraint generation method with each subproblem solved by the semismooth Newton based proximal augmented Lagrangian method (PALM) for the metric nearness problem. Due to the high memory requirement for the storage of the matrix related to the metric constraints, we take advantage of the special structure of the matrix and do not need to store the corresponding constraint matrix. A pleasing aspect of our algorithm is that we can solve these problems involving up to $10^{8}$ variables and $10^{13}$ constraints. Numerical experiments demonstrate the efficiency of our algorithm. In theory, firstly, under a mild condition, we establish a primal-dual error bound condition which is very essential for the analysis of local convergence rate of PALM. Secondly, we prove the equivalence between the dual nondegeneracy condition and nonsingularity of the generalized Jacobian for the inner subproblem of PALM. Thirdly, when $q(\cdot)=\|\cdot\|_{1}$ or $\|\cdot\|_{\infty}$, without the strict complementarity condition, we also prove the equivalence between the the dual nondegeneracy condition and the uniqueness of the primal solution.

Various problems in machine learning, databases, and statistics involve pairwise distances among a set of objects. It is often desirable for these distances to satisfy the properties of a metric, especially the triangle inequality. Applications where metric data is useful include clustering, classification, metric-based indexing, and approximation algorithms for various graph problems. This paper presents the Metric Nearness Problem: Given a dissimilarity matrix, find the "nearest" matrix of distances that satisfy the triangle inequalities.

Various problems in machine learning, databases, and statistics involve pairwise distances among a set of objects. It is often desirable for these distances to satisfy the properties of a metric, especially the triangle inequality. Applications where metric data is useful include clustering, classification, metric-based indexing, and approximation algorithms for various graph problems. This paper presents the Metric Nearness Problem: Given a dissimilarity matrix, find the "nearest" matrix of distances that satisfy the triangle inequalities.

Various problems in machine learning, databases, and statistics involve pairwise distances among a set of objects. It is often desirable for these distances to satisfy the properties of a metric, especially the triangle inequality. Applicationswhere metric data is useful include clustering, classification, metric-based indexing, and approximation algorithms for various graph problems. This paper presents the Metric Nearness Problem: Givena dissimilarity matrix, find the "nearest" matrix of distances that satisfy the triangle inequalities.