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Register a new userA general learning algorithm for functions between metric spaces
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Authors
Kerry M. Soileau
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In this paper we show how to approximate (learn) a function f, where X and Y are metric spaces.
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We prove a compactness result for bounded sequences $(u_j)_j$ of functions with bounded variation in metric spaces $(X,d_j)$ where the space $X$ is fixed but the metric may vary with $j$. We also provide an application to Carnot-Caratheodory spaces.
Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylors theorem come handy for this purpose. While the former is useful in estimating $|f(A+X)-f(A)|$ in terms of $|X|$ and requires the norms of the first derivative of the function, the latter is useful in computing higher order perturbation bounds and needs norms of the higher order derivatives of the function. In the study of matrices, determinant is an important function. Other scalar valued functions like eigenvalues and coefficients of characteristic polynomial are also well studied. Another interesting function of this category is the permanent, which is an analogue of the determinant in matrix theory. More generally, there are operator valued functions like tensor powers, antisymmetric tensor powers and symmetric tensor powers which have gained importance in the past. In this article, we give a survey of the recent work on the higher order derivatives of these functions and their norms. Using Taylors theorem, higher order perturbation bounds are obtained. Some of these results are very recent and their detailed proofs will appear elsewhere.
We show that given a homeomorphism $f:GrightarrowOmega$ where $G$ is a open subset of $mathbb{R}^2$ and $Omega$ is a open subset of a $2$-Ahlfors regular metric measure space supporting a weak $(1,1)$-Poincare inequality, it holds $fin BV_{operatorname{loc}}(G,Omega)$ if and only $f^{-1}in BV_{operatorname{loc}}(Omega,G)$. Further if $f$ satisfies the Luzin N and N$^{-1}$ conditions then $fin W^{1,1}_{operatorname{loc}}(G,Omega)$ if and only if $f^{-1}in W^{1,1}_{operatorname{loc}}(Omega,G)$.
We prove that every Banach space, not necessarily separable, can be isometrically embedded into a $mathcal L_{infty}$-space in a way that the corresponding quotient has the Radon-Nikodym and the Schur properties. As a consequence, we obtain $mathcal L_infty$ spaces of arbitrary large densities with the Schur and the Radon-Nikodym properties. This extents the a classical result by J. Bourgain and G. Pisier.
We characterize the (essentially) decreasing sequences of positive numbers $beta$ = ($beta$ n) for which all composition operators on H 2 ($beta$) are bounded, where H 2 ($beta$) is the space of analytic functions f in the unit disk such that $infty$ n=0 |c n | 2 $beta$ n < $infty$ if f (z) = $infty$ n=0 c n z n. We also give conditions for the boundedness when $beta$ is not assumed essentially decreasing.
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