Descent modulus and applications

The norm of the gradient $\\nabla$f (x) measures the maximum descent of a\nreal-valued smooth function f at x. For (nonsmooth) convex functions, this is\nexpressed by the distance dist(0, $\\partial$f (x)) of the subdifferential to\nthe origin, while for general real-valued functions defined on metric spaces by\nthe notion of metric slope |$\\nabla$f |(x). In this work we propose an\naxiomatic definition of descent modulus T [f ](x) of a real-valued function f\nat every point x, defined on a general (not necessarily metric) space. The\ndefinition encompasses all above instances as well as average descents for\nfunctions defined on probability spaces. We show that a large class of\nfunctions are completely determined by their descent modulus and corresponding\ncritical values. This result is already surprising in the smooth case: a\none-dimensional information (norm of the gradient) turns out to be almost as\npowerful as the knowledge of the full gradient mapping. In the nonsmooth case,\nthe key element for this determination result is the break of symmetry induced\nby a downhill orientation, in the spirit of the definition of the metric slope.\nThe particular case of functions defined on finite spaces is studied in the\nlast section. In this case, we obtain an explicit classification of descent\noperators that are, in some sense, typical.\n