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Wahba's classical representer theorem states that the solutions of certain risk minimization problems involving an empirical risk term and a quadratic regularizer can be written as expansions in terms of the training examples. We generalize the theorem to a larger class of regularizers

The representer theorem assures that kernel methods retain optimality under penalized empirical risk minimization. While a sufficient condition on the form of the regularizer guaranteeing the representer theorem has been known since the initial development of kernel methods, necessary conditions

The Brown representability theorem gives a list of conditions for the representablity of a set-valued contravariant functor which is defined on the classical stable homotopy category. It has had many uses through the years, and has

In this note, hyperseparoids are introduced; hyperseparoids are to separoids as Tverberg’s theorem is to Radon’s theorem. Also, a geometric representation theorem for acyclic k-separoids is presented which generalises that for separoids exhibited in [2].

The tetrad representation theorem, due to Spirtes, Glymour, and Scheines (1993), gives a graphical condition necessary and sufficient for the vanishing of tetrad differences in a linear correlation structure. This note simplifies their proof and generalizes the theorem. This generalization can

In the article the representation theorem for finite distributive lattice as rings of sets is presented. Auxiliary concepts are introduced. Namely, the concept of the height of an element, the maximal element in a chain, immediate predecessor of an element and ring of sets. Besides the scheme

Abstract. Existence of pramarts selectors for multivalued pramart who-se values are convex weakly compact subsets of a separable Banach space E (resp. subsets of a dual space E∗) are established. Representation theorems for multivalued pramarts are also presented. 1.

Abstract. In this paper we present a variant of the well known Skorokhod Representation Theorem. In our main result, given S a Polish Space, to a given continuous path α in the space of probability measures on S, we associate a continuous path in the space of S-valued random variables on a

We discuss proofs of representation theorems for implication structures, being algebraic frames for substructural logics, as e.g. the Lambek calculus and BCI (see Ono and Komori [9]). For product-free fragments of these logics, the theorems can be proven by methods essentially analogous to those