Geometry and Coarse-Grained Representations of Landscapes

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Basic geometric notions describing the structure of landscapes as well as the dynamics of local search on them include basins, saddles, reachability and funnels. We focus on discrete, combinatorial landscapes and emphasize the complications arising from local degeneracies. Local search in such landscapes is well described by adaptive walks, which we use to define reachability of a target from an initial configuration. Reachability introduces a topological structure on the configuration space. Combinatorial vector fields (CVFs) provide a more powerful mathematical framework in which the subtleties of local degeneracy can be conveniently formalized. Stochastic search dynamics has a direct representation as a probability space over the set of CVFs with the given landscape as a Lyapunov function. This ensemble of CVFs is amenable to the framework of standard statistical mechanics. The implications of landscape structure on search dynamics are elucidated further by the fact that the set of all CVFs on a landscape has a product structure, factorizing over extended plateaus (so called shelves) of the landscape. Finally, we discuss the coarse graining of landscapes from two perspectives. Traditionally, a partitioning (e.g. by gradient basins) of a given landscape is used to obtain a landscape with fewer configurations called macrostates. A reverse, and less investigated, view on coarse graining considers finer landscapes, with a larger number of configurations than the original one and a non-injective mapping into the original configuration space. Such encodings of landscapes, when suitably defined, turn out advantageous for optimization by adaptive walks.