Aggregation of multiple metric descriptions from distances between unlabeled objects

• Published in: Computational mathematics and mathematical physics Vol. 57; no. 2; pp. 350 - 361
• Main Authors: Maysuradze, A. I., Suvorov, M. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.02.2017; Springer Nature B.V
• Subjects: Agglomeration; Algorithms; Approximation; Computational mathematics; Computational Mathematics and Numerical Analysis; Descriptions; Factorization; Financial analysis; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix; Methods; Object recognition; Optimization; Position tracking; Principal components analysis; Studies; United States > US; multiple metric spaces; similarity measures; nonnegative matrix factorization (NMF); multiple metric descriptions; dimension reduction; principal component analysis (PCA)
• ISSN: 0965-5425; 1555-6662
• DOI: 10.1134/S0965542517020105

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.