Unilateral analysis, orientation and determination

Abstract

The norm of the gradient ‖∇𝑓(𝑥)‖ measures the maximum descent of a smooth function 𝑓 at 𝑥. For nonsmooth convex functions, this is expressed by the remoteness of the subdifferential (that is, the distance of 𝜕𝑓(𝑥) to the origin), while for general real-valued functions defined on metric spaces by the notion of metric slope 𝑠𝑓(𝑥) due to De Giorgi. More generally, an axiomatic definition of descent modulus is possible, for functions defined on general spaces (not necessarily metric), encompassing both the (metric) notion of steepest descent as well as the (probabilistic) notion of average descent for functions defined on probability spaces. A large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope.