distancederived

Distancederived is a term used in mathematics and data analysis to describe quantities, features, or metrics that are computed exclusively from pairwise distances among a set of objects. In this usage, a distancederived quantity depends only on the distance matrix d(x_i, x_j) and is typically invariant under rigid motions and reordering of the objects.

From a formal perspective, given a finite set X with a distance function d, a distancederived feature F maps the distance matrix D to a real value, often with permutation invariance F(x1,...,xn) = F(pi(x1),...,pi(xn)) for any permutation pi. Examples include the mean or variance of pairwise distances, the full distance distribution, or distance-based kernels such as k(d) = exp(-gamma d^2). In a graph formulation, a distancederived object may be a Laplacian or adjacency structure built from distances, whose spectrum or eigenvectors yield features.

Applications of distancederived quantities appear in clustering, dimensionality reduction, anomaly detection, and shape comparison, especially when coordinate representations are unavailable or unreliable. They also support graph construction methods (for example, building k-nearest-neighbor graphs from distances) and enable global structural analysis of point sets.

Limitations include sensitivity to scale and noise, potential computational cost for large datasets, and possible loss of information about absolute position or orientation. Distancederived features are typically used alongside coordinate-based features when those are accessible.

See also: metric spaces, distance metrics, kernel methods, multidimensional scaling, graph Laplacians.