Optimal partitions and Semi-discrete optimal transport

In the current book I suggest an off-road path to the subject of optimal transport. I tried to avoid prior knowledge of analysis, PDE theory and functional analysis, as much as possible. Thus I concentrate on discrete and semi-discrete cases, and always assume compactness for the underlying spaces. However, some fundamental knowledge of measure theory and convexity is unavoidable. In order to make it as self-contained as possible I included an appendix with some basic definitions and results. I believe that any graduate student in mathematics, as well as advanced undergraduate students, can read and understand this book. Some chapters (in particular in Parts II\&III ) can also be interesting for experts. Starting with the the most fundamental, fully discrete problem I attempted to place optimal transport as a particular case of the celebrated stable marriage problem. From there we proceed to the partition problem, which can be formulated as a transport from a continuous space to a discrete one. Applications to information theory and game theory (cooperative and non-cooperative) are introduced as well. Finally, the general case of transport between two compact measure spaces is introduced as a coupling between two semi-discrete transports.

[1]  H. Joe.  Majorization, randomness and dependence for multivariate distributions , 1987 .

[2]  Paul L. Zador,et al.  Asymptotic quantization error of continuous signals and the quantization dimension , 1982, IEEE Trans. Inf. Theory.

[3]  Naftali Tishby,et al.  The information bottleneck method , 2000, ArXiv.

[4]  Tryphon T. Georgiou,et al.  On Matrix-Valued Monge–Kantorovich Optimal Mass Transport , 2015, IEEE Transactions on Automatic Control.

[5]  Gershon Wolansky.  On Semi-discrete Monge–Kantorovich and Generalized Partitions , 2015, J. Optim. Theory Appl..

[6]  S. Rachev,et al.  Mass transportation problems , 1998 .

[7]  C. Villani.  Topics in Optimal Transportation , 2003 .

[8]  H. Scarf.  The Core of an N Person Game , 1967 .

[9]  A. Vershik.  Long History of the Monge-Kantorovich Transportation Problem , 2013 .

[10]  On displacement interpolation of measures involved in Brenier’s theorem , 2011 .

[11]  Tryphon T. Georgiou,et al.  Signals and control aspects of optimal mass transport and the Boltzmann entropy , 2010, 49th IEEE Conference on Decision and Control (CDC).

[12]  F. Santambrogio.  Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling , 2015 .

[13]  Gershon Wolansky,et al.  On optimal partitions, individual values and cooperative games: Does a wiser agent always produce a higher value? , 2015, 1506.01242.

[14]  E. Maskin.  Nash Equilibrium and Welfare Optimality , 1999 .

[15]  Tryphon T. Georgiou,et al.  Distances and Riemannian Metrics for Multivariate Spectral Densities , 2011, IEEE Transactions on Automatic Control.

[16]  H. Moulin.  Cooperative Microeconomics: A Game-Theoretic Introduction , 1995 .

[17]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[18]  Harry Joe,et al.  Majorization and divergence , 1990 .

[19]  R. Schneider.  Measures in Convex Geometry , 1998 .

[20]  Lloyd S. Shapley,et al.  On balanced sets and cores , 1967 .