In this paper we extend the definition of K-positive definite operators from linear to Fréchet differentiable operators. Under this setting, we derive from the inverse function theorem a local existence and approximation results corresponding to those of Theorems 1 and 2 of the authors [8], in an arbitrary real Banach space. Furthermore, an asymptotically K-positive definite operator is introdu...

Several attempts have been made to modify the quasi-Newton condition in order obtain rapid convergence with complete properties (symmetric and positive definite) of inverse Hessian matrix (second derivative objective function). There are many unconstrained optimization methods that do not generate definiteness matrix. One those is symmetric rank 1( H-version) update (SR1 update), where this sat...

This document contains detailed proofs of theorems stated in the main article entitled Random Feature Maps for Dot Product Kernels. 1 Proof of Theorem 1 We first recollect Schoenberg’s result in its original form Theorem 1 (Schoenberg (1942), Theorem 2). A function f : [−1, 1]→ R constitutes a positive definite kernel K : S∞ × S∞ → R, K : (x,y) 7→ f(〈x,y〉) iff f is an analytic function admittin...