A recovering of violated metric in machine learning

Experimental results in machine learning, data analysis and data mining often appear as comparisons between elements from a limited set. If a matrix of pairwise similarities is positively definite, then the set of elements is considered to be immersed in some metric space, e.g. Euclidean one, with dimensionality no more than the rank of the matrix. But it can be the non-positively definite matrix, because measurement results usually are not scalar products. It is necessary to recover metric for correct use of it in clustering or machine learning problems. Violations arise not only in the triangle inequality, but also relative to positions of more than three elements. In general, all similarity submatrices for triples of elements are positively definite, but simultaneously the whole matrix is negatively definite. We discuss here the approach to recover violated metric based on the idea of appropriate corrections of normalized similarity matrix and develop it for non-normalized similarity and dissimilarity ones.