%0 Book Section %1 statphys23_0493 %A Yasuda, M. %A Tanaka, K. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K expansion ising learning linear machine mean-field plefka random response spin statphys23 theory topic-11 %T The approximate linear response theory and its mathmatical structures %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=493 %X Random Ising models are useful for probabilistic information processing (PIP) as well as for statistical mechanics1,2. Recently, the models have been applied to image processing, error correcting code theory, code-division multiple-access (CDMA) multi-user detection in wireless telecommunication, machine learning and other various problems2-6. In these applications, correlation functions of two spins are often important as well as thermal averages of one spin. Plefka's expansion7 is one of the most important approximate method in the mean-field theory, and the relationship between this method and the cluster variation method has been pointed out8. However, understanding how correlation functions are calculated by using Plefka's expansion is nontrivial. Several authors proposed effective approximate method to calculate correlation functions of two spins by combining the mean-field theory based on Plefka's expansion with the linear response theory9. However, it is not clear how the approximate correlations functions obtained by the above method approximate true correlation functions. We are interested in understanding how correlation functions are calculated by their method and the reasons why it leads to good results. We will explain these two topics using perturbative point of view. \subsubsection*Reference łabelenumi þeenumi enumerate ıtem H. Nishimori: Statistical Physics of Spin Glass and Information Processing --- an Introduction (Oxford University Press, Oxford, 2001). ıtem M. Opper and D. Saad: Advanced Mean Field Methods --- Theory and Practice (Cambridge, MA: MIT Press, 2001). ıtem K. Tanaka: J. Phys. A, 35 (2002) R81. ıtem Y. Kabashima and D. Saad: J. Phys. A. Math. Gen., 37 (2004) R1. ıtem T. Tanaka: IEEE Trans. Inform. Theory, 48 (2002) 2888. ıtem M. Yasuda and T. Horiguchi: Physca A, 368 (2006) 83. ıtem T. Plefka: J. Phys. A, 15 (1982) 1971. ıtem M. Yasuda and K. Tanaka: J. Phys. Soc. Jpn., 75 (2006) 1. ıtem M. A. R. Leisink and H. J. Kappen: Neural Networks, 13 (2000) 329. enumerate

@incollection{statphys23_0493, abstract = {Random Ising models are useful for probabilistic information processing (PIP) as well as for statistical mechanics[1,2]. Recently, the models have been applied to image processing, error correcting code theory, code-division multiple-access (CDMA) multi-user detection in wireless telecommunication, machine learning and other various problems[2-6]. In these applications, correlation functions of two spins are often important as well as thermal averages of one spin. Plefka's expansion[7] is one of the most important approximate method in the mean-field theory, and the relationship between this method and the cluster variation method has been pointed out[8]. However, understanding how correlation functions are calculated by using Plefka's expansion is nontrivial. Several authors proposed effective approximate method to calculate correlation functions of two spins by combining the mean-field theory based on Plefka's expansion with the linear response theory[9]. However, it is not clear how the approximate correlations functions obtained by the above method approximate true correlation functions. We are interested in understanding how correlation functions are calculated by their method and the reasons why it leads to good results. We will explain these two topics using perturbative point of view. \subsubsection*{Reference} \renewcommand{\labelenumi}{ [\theenumi] } \begin{enumerate} \item H. Nishimori: {\textit{Statistical Physics of Spin Glass and Information Processing --- an Introduction}} (Oxford University Press, Oxford, 2001). \item M. Opper and D. Saad: {\textit{Advanced Mean Field Methods --- Theory and Practice}} (Cambridge, MA: MIT Press, 2001). \item K. Tanaka: J. Phys. A, {\textbf{35}} (2002) R81. \item Y. Kabashima and D. Saad: J. Phys. A. Math. Gen., {\textbf{37}} (2004) R1. \item T. Tanaka: IEEE Trans. Inform. Theory, {\textbf{48}} (2002) 2888. \item M. Yasuda and T. Horiguchi: Physca A, {\textbf{368}} (2006) 83. \item T. Plefka: J. Phys. A, {\textbf{15}} (1982) 1971. \item M. Yasuda and K. Tanaka: J. Phys. Soc. Jpn., {\textbf{75}} (2006) 1. \item M. A. R. Leisink and H. J. Kappen: Neural Networks, {\textbf{13}} (2000) 329. \end{enumerate}}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Yasuda, M. and Tanaka, K.}, biburl = {https://www.bibsonomy.org/bibtex/2ae13b90d6262673549b5fc52affe7268/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {322b3bbd70704c4b77813fa99c2f18bc}, intrahash = {ae13b90d6262673549b5fc52affe7268}, keywords = {expansion ising learning linear machine mean-field plefka random response spin statphys23 theory topic-11}, month = {9-13 July}, timestamp = {2007-06-20T10:16:21.000+0200}, title = {The approximate linear response theory and its mathmatical structures}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=493}, year = 2007 }