A coordinated ranking as an opinion of an expert group can be represented by the well-known Kemeny’s median. Such a decision is a least different from other rankings, and is free of some ...contradictions of the majority rule problem. As a mathematical principle, the Kemeny’s median gives a decision in any case, in particular, for conflicting experts or groups. In practice, competing opinions are usually identified, and special procedures are used to achieve the required level of consensus. One known approach consists in assigning weights to expert opinions. In this paper, the correct mathematical basis is formulated for new types on the Kemeny’s median, like metric and weighted ones. The problem to find the median for a linear combination of expert rankings is investigated based on the well-known locally optimal Kemeny’s algorithm. It is proposed to investigate the rank aggregation problem based on new types of the median.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Recently, raw experimental data in machine learning often appear as direct comparisons between objects (featureless data). Different ways to evaluate difference or similarity of a pair of objects in ...image and data mining, image analysis, bioinformatics, etc., are usually used in practice. Nevertheless, such comparisons often are not distances or correlations (scalar products) like a correct function defined on a limited set of elements in machine learning. This problem is denoted as metric violations in ill-posed matrices. Therefore, it needs to recover violated metrics and provide optimal conditionality of corresponding matrices of pairwise comparisons for distances and similarities. This is the correct basis for using of modern machine learning algorithms.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
A coordinated ranking as the opinion of an expert group usually can be represented by the well-known Kemeny's median. The Kemeny's median is the least different ranking from other rankings and is ...free of known contradictions of the majority rule problem. As a mathematical principle, the Kemeny's median gives a decision in any case, in particular, for conflicting experts’ decisions in ordinal scales. In practice, competing opinions are usually modified by special approval procedures to achieve the required level of consensus. The known approach consists in assigning weights to experts’ opinions. In this paper, the problem to find the median for a linear combination of experts’ rankings is investigated using the well-known locally optimal Kemeny's algorithm. It is proposed to use the weighted loss matrix in it.
A recovering of violated metric in machine learning Dvoenko, Sergey D.; Pshenichny, Denis O.
Proceedings of the Seventh Symposium on Information and Communication Technology,
12/2016
Conference Proceeding
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.
