Dualistic Riemannian Manifold Structure Induced from Convex Functions
Convex analysis has wide applications in science and engineering, such as mechanics, optimization and control, theoretical statistics, mathematical economics and game theory, and so on. It offers an analytic framework to treat systems and phenomena that depart from linearity, based on an elegant mathematical characterization of the notion of “duality” (Rockafellar, 1970, 1974, Ekeland and Temam, 1976). Recent work of David Gao (2000) further provided a comprehensive and unified treatment of duality principles in convex and nonconvex systems, greatly enriching the theoretical foundation and scope of applications.
Key wordsLegendre–Fenchel duality biorthogonal coordinates Riemannian metric conjugate connections equiaffine geometry parallel volume form affine immersion Hessian geometry
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- Amari, S. (1985). Differential Geometric Methods in Statistics. Lecture Notes in Statistics 28, Springer-Verlag, New York. Reprinted in 1990.Google Scholar
- Amari, S. and Nagaoka, H. (2000). Method of Information Geometry. AMS Monograph, Oxford University Press.Google Scholar
- Della Pietra, S., Della Pietra, V., and Lafferty, J. (2002). Duality and auxiliary functions for Bregman distances. Technical Report CMU-CS-01-109, School of Computer Science, Carnegie Mellon University.Google Scholar
- Gao, D.Y. (2000). Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer Academic, Dordrecht, xviii+454 pp.Google Scholar
- Nomizu, K. and Sasaki, T. (1994). Affine Differential Geometry – Geometry of Affine Immersions. Cambridge University Press.Google Scholar
- Rockafellar, R.T. (1970). Convex Analysis. Princeton University Press.Google Scholar
- Simon, U., Schwenk-Schellschmidt, A., and Viesel, H. (1991). Introduction to the Affine Differential Geometry of Hypersurfaces. Lecture Notes, Science University of Tokyo.Google Scholar
- Zhang, J. (2006a). Referential duality and representational duality in the scaling of multidimensional and infinite-dimensional stimulus space. In: Dzhafarov, E. and Colonius, H. (Eds.) Measurement and Representation of Sensations: Recent Progress in Psychological Theory. Lawrence Erlbaum, Mahwah, NJ.Google Scholar
- Zhang, J. (2006b). Referential duality and representational duality on statistical manifolds. Proceedings of the Second International Symposium on Information Geometry and Its Applications, Tokyo (pp. 58–67).Google Scholar