There exists a calculus for general nondifferentiable functions that englobes a large part of the familiar subdifferential calculus for convex nondifferentiable functions [1]. This development started with F.H. Clarke, who introduced a generalized gradient for functions that are locally Lipschitz, but (possibly) nondifferentiable. Generalized gradients turn out to be the subdifferentials, in th...

This work studies the typical behavior of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M -Lipschitz functions (functions which change by at most M along edges) and integer-homomorphisms (functions which change by exactly 1 along edges). We prove that such functions typically exhibit very small fluctuations....

The goal of the paper is to design sequential strategies which lead to efficient optimization of an unknown function under the only assumption that it has a finite Lipschitz constant. We first identify sufficient conditions for the consistency of generic sequential algorithms and formulate the expected minimax rate for their performance. We introduce and analyze a first algorithm called LIPO wh...

We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can be decomposed f into a finite number of BiLipschitz functions f |Fi so that the k-Hausdorff content of f([0, 1] r ∪Fi) is small. We thus generalize a theorem of P. Jones [Jon88] from the setting of R to the setting of a general metric spa...

In this paper we address some of the most fundamental questions regarding the diierentiability structure of locally Lipschitz functions deened on Banach spaces. For example, we examine the relationship between inte-grability, D-representability and strict diierentiability. In addition to this, we show that on a large class of Banach spaces there is a signiicant family of locally Lipschitz funct...

We prove that if f is a Lipschitz function on R, A and B are self-adjoint operators such that rank(A − B) = 1, then f(A) − f(B) belongs to the weak space S1,∞, i.e., sj(A − B) ≤ const(1 + j). We deduce from this result that if A − B belongs to the trace class S1 and f is Lipschitz, then f(A) − f(B) ∈ SΩ, i.e., Pn j=0 sj(f(A) − f(B)) ≤ const log(2 + n). We also obtain more general results about ...

We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. We study the relations of such functions with topometric versions of classical separation axioms, namely, normality and complete regularity, as well as with completions of topometric spaces. We ...

The goal of this article is to develop a framework for large margin classification in metric spaces. We want to find a generalization of linear decision functions for metric spaces and define a corresponding notion of margin such that the decision function separates the training points with a large margin. It will turn out that using Lipschitz functions as decision functions, the inverse of the...

in this paper, using a generalized dunkl translation operator, we obtain an analog of titchmarsh's theorem for the dunkl transform for functions satisfying the lipschitz-dunkl condition in $mathrm{l}_{2,alpha}=mathrm{l}_{alpha}^{2}(mathbb{r})=mathrm{l}^{2}(mathbb{r}, |x|^{2alpha+1}dx), alpha>frac{-1}{2}$.