application of graph

application of graph

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General Statistical Models

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Solution of a linear system

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  Solution of a linear system [ edit ] The steepest descent algorithm applied to the  Wiener filter [11] Gradient descent can be used to solve a system of linear equations � � − � = 0 reformulated as a quadratic minimization problem. If the system matrix  �  is real  symmetric  and  positive-definite , an objective function is defined as the quadratic function, with minimization of � ( � ) = � � � � − 2 � � � , so that ∇ � ( � ) = 2 ( � � − � ) . For a general real matrix  � ,  linear least squares  define � ( � ) = ‖ � � − � ‖ 2 . In traditional linear least squares for real  �  and  �  the  Euclidean norm  is used, in which case ∇ � ( � ) = 2 � � ( � � − � ) . The  line search  minimization, finding the locally optimal step size  �  on every iteration, can be performed analytically for quadratic functions, and explicit formulas for the locally optimal  �  are known. [5] [12] For example, for real  symmetric  and  positive-definite  matrix  � , a simple algorithm can be as follows, [5
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