Aggregation of multiple metric descriptions from distances between unlabeled objects

The situation when there are several different semimetrics on the set of objects in the recognition problem is considered. The problem of aggregating distances based on an unlabeled sample is stated and investigated. In other words, the problem of unsupervised reduction of the dimension of multiple metric descriptions is considered. This problem is reduced to the approximation of the original distances in the form of optimal matrix factorization subject to additional metric constraints. It is proposed to solve this problem exactly using the metric nonnegative matrix factorization. In terms of the problem statement and solution procedure, the metric data method is an analog of the principal component method for feature-oriented descriptions. It is proved that the addition of metric requirements does not decrease the quality of approximation. The operation of the method is demonstrated using toy and real-life examples.