International audience

Optimal transport was originally invented by Gaspard Monge [“Mémoire sur la théorie des déblais et des remblais,” Mem. Math. Phys. Acad. Royale Sci., (1781), pp. 666–704] to model the problem of optimally mapping one distribution of mass onto another. This was later reformulated by Leonid Kantorovich as a well-posed linear program using the notion of transport plans instead of maps in [“On the translocation of masses,” Dokl. Akad. Nauk. USSR (N.S.), 37 (1942), pp. 199–201], which earned him the Nobel Memorial Prize in Economic Sciences. In the past four decades the field of optimal transport has grown far beyond its original purpose and has evolved into a driving force for applications both within mathematics and in other sciences. This book review deals with the new monograph Optimal Mass Transport on Euclidean Spaces by Francesco Maggi.

A dynamical system is a transformation of a phase space, and the transformation law is the primary means of defining as well as identifying the dynamical system and is the object of focus of many learning techniques. However, there are many secondary aspects of dynamical systems—invariant sets, the Koopman operator, and Markov approximations—that provide alternative objectives for learning techniques. Crucially, while many learning methods are focused on the transformation law, we find that forecast performance can depend on how well these other aspects of the dynamics are approximated. These different facets of a dynamical system correspond to objects in completely different spaces—namely, interpolation spaces, compact Hausdorff sets, unitary operators, and Markov operators, respectively. Thus, learning techniques targeting any of these four facets perform different kinds of approximations. We examine whether an approximation of any one of these aspects of the dynamics could lead to an approximation of another facet. Many connections and obstructions are brought to light in this analysis. Special focus is placed on methods of learning the primary feature—the dynamics law itself. The main question considered is the connection between learning this law and reconstructing the Koopman operator and the invariant set. The answers are tied to the ergodic and topological properties of the dynamics, and they reveal how these properties determine the limits of forecasting techniques.