. . . "1099751081"^^ . . . . . . . . "\u0414\u0438\u0432\u0435\u0440\u0433\u0435\u043D\u0446\u0438\u044F \u0411\u0440\u044D\u0433\u043C\u0430\u043D\u0430"@ru . . . . . . . . . "yes"@en . "4491248"^^ . . . . . . . . . . . "InternetArchiveBot"@en . . . . . . . . . "Divergence de Bregman"@fr . . . . . . . . . . . . . . . . . . . . . "Bregman divergence"@en . . . . . . . . . . . "thumb|Bregman divergence interpreted as areas.\nFor any , define for . Let .\n\nThen for , and since is continuous, also for .\n\nThen, from the diagram, we see that for for all , we must have linear on .\n\nThus we find that varies linearly along any direction. By the next lemma, is quadratic. Since is also strictly convex, it is of form , where .\n\nLemma: If is an open subset of , has continuous derivative, and given any line segment , the function is linear in , then is a quadratic function.\n\nProof idea: For any quadratic function , we have still has such derivative-linearity, so we will subtract away a few quadratic functions and show that becomes zero.\n\nThe proof idea can be illustrated fully for the case of , so we prove it in this case.\n\nBy the derivative-linearity, is a quadratic function on any line segment in . We subtract away four quadratic functions, such that becomes identically zero on the x-axis, y-axis, and the line.\n\nLet , for well-chosen . Now use to remove the linear term, and use respectively to remove the quadratic terms along the three lines.\n\n not on the origin, there exists a line across that intersects the x-axis, y-axis, and the line at three different points. Since is quadratic on , and is zero on three different points, is identically zero on , thus . Thus is quadratic."@en . . "In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted as probability distributions \u2013 notably as either values of the parameter of a parametric model or as a data set of observed values \u2013 the resulting distance is a statistical distance. The most basic Bregman divergence is the squared Euclidean distance. Bregman divergences are similar to metrics, but satisfy neither the triangle inequality (ever) nor symmetry (in general). However, they satisfy a generalization of the Pythagorean theorem, and in information geometry the corresponding statistical manifold is interpreted as a (dually) flat manifold. This allows many techniques of optimization theory to be generalized to Bregman divergences, geometrically as generalizations of least squares. Bregman divergences are named after Russian mathematician Lev M. Bregman, who introduced the concept in 1967."@en . . . "In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted as probability distributions \u2013 notably as either values of the parameter of a parametric model or as a data set of observed values \u2013 the resulting distance is a statistical distance. The most basic Bregman divergence is the squared Euclidean distance."@en . "June 2019"@en . . . . . . . . . "En math\u00E9matiques, la divergence de Bregman est une mesure de la diff\u00E9rence entre deux distributions d\u00E9riv\u00E9e d'une fonction potentiel U \u00E0 valeurs r\u00E9elles strictement convexe et contin\u00FBment diff\u00E9rentiable. Le concept a \u00E9t\u00E9 introduit par (en) en 1967.Par l'interm\u00E9diaire de la transformation de Legendre, au potentiel correspond un potentiel dual et leur diff\u00E9rentiation donne naissance \u00E0 deux syst\u00E8mes de coordonn\u00E9es duaux."@fr . . . . "24956"^^ . . . . . . . . . "hidden"@en . . . . . . 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"En math\u00E9matiques, la divergence de Bregman est une mesure de la diff\u00E9rence entre deux distributions d\u00E9riv\u00E9e d'une fonction potentiel U \u00E0 valeurs r\u00E9elles strictement convexe et contin\u00FBment diff\u00E9rentiable. Le concept a \u00E9t\u00E9 introduit par (en) en 1967.Par l'interm\u00E9diaire de la transformation de Legendre, au potentiel correspond un potentiel dual et leur diff\u00E9rentiation donne naissance \u00E0 deux syst\u00E8mes de coordonn\u00E9es duaux."@fr . . 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