Generalized vs Set Median String for Histogram Based Distances: Algorithms and Classification Results in the Image Domain

We compare different statistical characterizations of a set of strings, for three different histogram-based distances. Given a distance, a set of strings may be characterized by its generalized median, i.e., the string —over the set of all possible strings— that minimizes the sum of distances to every string of the set, or by its set median, i.e., the string of the set that minimizes the sum of distances to every other string of the set. For the first two histogram-based distances, we show that the generalized median string can be computed efficiently; for the third one, which biased histograms with individual substitution costs, we conjecture that this is a NP-hard problem, and we introduce two different heuristic algorithms for approximating it. We experimentally compare the relevance of the three histogram-based distances, and the different statistical characterizations of sets of strings, for classifying images that are represented by strings.