We consider the problem of learning on a compact metric space X in a functional analytic framework. For a dense subalgebra of Lip(X), the space of all Lipschitz functions on X, the Representer Theorem is derived. We obtain exact solutions in the case of least square minimization and regularization and suggest an approximate solution for the Lipschitz classifier.

(1.1) |f(a)− f(b)| ≤ L |a− b| for every pair of points a, b ∈ A. We also say that a function is Lipschitz if it is L-Lipschitz for some L. The Lipschitz condition as given in (1.1) is a purely metric condition; it makes sense for functions from one metric space to another. In these lectures, we concentrate on the theory of Lipschitz functions in Euclidean spaces. In Section 2, we study extensio...

in this paper, we introduce the concepts of $2$-isometry, collinearity, $2$%-lipschitz mapping in $2$-fuzzy $2$-normed linear spaces. also, we give anew generalization of the mazur-ulam theorem when $x$ is a $2$-fuzzy $2$%-normed linear space or $im (x)$ is a fuzzy $2$-normed linear space, thatis, the mazur-ulam theorem holds, when the $2$-isometry mapped to a $2$%-fuzzy $2$-normed linear space...

Existence and uniqueness of complex geodesics joining two points of a convex bounded domain in a Banach space X are considered. Existence is proved for the unit ball of X under the assumption that X is 1-complemented in its double dual. Another existence result for taut domains is also proved. Uniqueness is proved for strictly convex bounded domains in spaces with the analytic Radon-Nikodym pro...

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...

It is well known that the absolute value map on the self-adjoint operators on an infinite dimensional Hilbert spaces is not Lipschitz continuous, although Lipschitz continuity holds on certain subsets of operators. In this note, we provide an elementary proof that the absolute value map is Lipschitz continuous on the set of all operators which are bounded below in norm by any fixed positive con...

We develop a theory of removable singularities for the weighted Bergman space Aμ(Ω) = {f analytic in Ω : R Ω |f | dμ < ∞}, where μ is a Radon measure on C. The set A is weakly removable for Aμ(Ω \ A) if Aμ(Ω \ A) ⊂ Hol(Ω), and strongly removable for Aμ(Ω \A) if Aμ(Ω \A) = Aμ(Ω). The general theory developed is in many ways similar to the theory of removable singularities for Hardy H spaces, BMO...

Lipschitz condition is a natural notion of function regularity in this context, and the norm dual to the mixed Lipschitz space is a natural distance between measures. In this paper, we consider the tensor product of spaces equipped with tree metrics and give effective formulas for the mixed Lipschitz norm and its dual. We also show that these norms behave well when approximating an arbitrary me...

In this paper we show that the study of integrability and D-representability of Lipschitz functions deened on arbitrary Banach spaces reduces to the study of these properties on separable Banach spaces.

It is shown that the category of non-Archimedean metric spaces with l-Lipschitz maps can be embedded as a coreflectlve non-bireflective subcategory in the category of fuzzy uniform spaces. Consequential characterizations of topological and unif’orm properties are derived.