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Statements

Subject Item
dbr:Bregman_divergence
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Дивергенция Брэгмана Divergence de Bregman Bregman divergence
rdfs:comment
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 – notably as either values of the parameter of a parametric model or as a data set of observed values – the resulting distance is a statistical distance. The most basic Bregman divergence is the squared Euclidean distance. Дивергенция Брэгмана или расстояние Брэгмана — мера расстояния между двумя точками, определённая в терминах строго выпуклой функции. Они образуют важный класс дивергенций. Если точки интерпретировать как распределение вероятностей, либо как значения , либо как набор наблюдаемых значений, то полученное расстояние является . Самой элементарной дивергенцией Брэгмана является квадрат евклидова расстояния. Дивергенция Брэгмана названа по имени Льва Мееровича Брэгмана, предложившего концепцию в 1967 году. En mathématiques, la divergence de Bregman est une mesure de la différence entre deux distributions dérivée d'une fonction potentiel U à valeurs réelles strictement convexe et continûment différentiable. Le concept a été introduit par (en) en 1967.Par l'intermédiaire de la transformation de Legendre, au potentiel correspond un potentiel dual et leur différentiation donne naissance à deux systèmes de coordonnées duaux.
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dbp:proof
thumb|Bregman divergence interpreted as areas. For any , define for . Let . Then for , and since is continuous, also for . Then, from the diagram, we see that for for all , we must have linear on . Thus we find that varies linearly along any direction. By the next lemma, is quadratic. Since is also strictly convex, it is of form , where . Lemma: If is an open subset of , has continuous derivative, and given any line segment , the function is linear in , then is a quadratic function. Proof 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. The proof idea can be illustrated fully for the case of , so we prove it in this case. By 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. Let , for well-chosen . Now use to remove the linear term, and use respectively to remove the quadratic terms along the three lines. 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.
dbp:drop
hidden
dbo:abstract
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 – notably as either values of the parameter of a parametric model or as a data set of observed values – 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 mathématiques, la divergence de Bregman est une mesure de la différence entre deux distributions dérivée d'une fonction potentiel U à valeurs réelles strictement convexe et continûment différentiable. Le concept a été introduit par (en) en 1967.Par l'intermédiaire de la transformation de Legendre, au potentiel correspond un potentiel dual et leur différentiation donne naissance à deux systèmes de coordonnées duaux. Дивергенция Брэгмана или расстояние Брэгмана — мера расстояния между двумя точками, определённая в терминах строго выпуклой функции. Они образуют важный класс дивергенций. Если точки интерпретировать как распределение вероятностей, либо как значения , либо как набор наблюдаемых значений, то полученное расстояние является . Самой элементарной дивергенцией Брэгмана является квадрат евклидова расстояния. Дивергенции Брэгмана подобны метрикам, но не удовлетворяют ни неравенству треугольника, ни симметрии (в общем случае), однако они удовлетворяют обобщённой теореме Пифагора. В соответствующее интерпретируется как (или двойственное). Это позволяет обобщить многие техники оптимизации к дивергенции Брэгмана, что геометрически соответствует обобщению метода наименьших квадратов. Дивергенция Брэгмана названа по имени Льва Мееровича Брэгмана, предложившего концепцию в 1967 году.
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