Multi-class Classication in Big Data

The paper suggests the on-line multi-class classier with a sublinear computational complexity relative to the number of training objects. The proposed approach is based on the combining of two-class probabilistic classifiers. Pairwise coupling is a popular multi-class classification method that combines all comparisons for each pair of classes. Unfortunately pairwise coupling suffers in many cases from incompatibility in that some regions of its input space the sum of probabilities are not equal to one. In this paper we propose the optimal approximation for probabilities in each point of object space. This paper proposes a new probabilistic interpretation of the Support Vector Machine for obtaining class probabilities. We show how the SVM can be viewed as a maximum likelihood estimate of a class of probabilistic models. As a computational method for big data we use the stochastic gradient descent approach minimizing directly the primal SVM objective. Unfortunately the hinge loss of the true SVM classier did not allow to use SGD procedure for determining the classier bias. In this paper we propose the piece-wise quadratic loss that helps to overcome this obstacle and gives an instrument to obtain the bias from SGD procedure

Pseudorutile-leucoxene-quartz ores of Timan ‒ a new genetic type of titanium raw materials: prospects for industrial development

The two largest deposits of Russia – Yaregskoye and Pizhemskoye belong to the same genetic type; hydrothermal-metamorphic indigenous deposits. They are located in the same Timan structure at a distance of no more than 230 km from each other. According to the total approved reserves and forecast resources of titanium dioxide, they are approaching 60% of the all-Russian and will form the basis of industrial titanium raw materials used in Russia in the near future. In the interests of technological mineralogy, morphological features, internal structure, chemical composition of grains of the two main titanium mineral phases ‒ leucoxene and pseudorutile, TiO2 polymorphs, as well as the composition of mineral microinclusions in these phases have been studied in detail. The compositions of all mineral phases in polished preparations of leucoxene and pseudorutile were analyzed by SEM-EDS method at the Institute of Geology and Geochronology of the Precambrian of the RAS, 147 chemical analyses were obtained at the point (3 µk) and many images of polished grains of anatase, leucoxene and pseudorutile were scanned over the area (20×20 µk). In the leucoxene grains themselves, 12 mineral phases were diagnosed and characterized in the form of inclusions: pseudorutile, rutile, anatase, quartz, hydromuscovite-illite, kaolinite, siderite, zircon, xenotime, pyrite, florencite, monazite and kularite. TiO2 polymorphs are verified by Raman spectroscopy and X-ray diffraction analysis. New evidence has been obtained that the transformation of ilmenite into leucoxene occurs hydrothermally through intermediate phases ‒ Fe-rutile and pseudorutile; the enlargement of rutile crystals in the leucoxene grain itself is shown; the presence of secondary crystals of siderite, florencite and others inside the studied grains