A note on range-kernel uncomplementation

A note on the Path Kernel Conjecture

Let τ(G) denote the number of vertices in a longest path in a graph G = (V , E). A subset K of V is called a Pn-kernel of G if τ(G[K ]) ≤ n−1 and every vertex v ∈ V \K is adjacent to an end-vertex of a path of order n− 1 in G[K ]. It is known that every graph has a Pn-kernel for every positive integer n ≤ 9. R. Aldred and C. Thomassen in [R.E.L. Aldred, C. Thomassen, Graphs with not all possibl...

A Note on Range-Restricted Circuit Covers

A Note on the Heat Kernel on the Heisenberg Group

Let ps be the convolution kernel of the operator e −sL (see [5, (1.10), (1.11)]). When s > 0, e−sL is the solution operator for the Heisenberg heat equation ∂su = −Lu and ps is called the heat kernel (see [6, (7.30), p. 71]. The goal of this note is to study the analytic continuation of the heat kernel ps. This is interesting from the point of view of the theory of analytic hypoellipticity (see...

A Note on Improved Loss Bounds for Multiple Kernel Learning

The paper [5] presented a bound on the generalisation error of classifiers learned through multiple kernel learning. The bound has (an improved) additive dependence on the number of kernels (with the same logarithmic dependence on this number). However, parts of the proof were incorrectly presented in that paper. This note remedies this weakness by restating the problem and giving a detailed pr...

A Note on Robust Kernel Principal Component Analysis

Extending the classical principal component analysis (PCA), the kernel PCA (Schölkopf, Smola and Müller, 1998) effectively extracts nonlinear structures of high dimensional data. But similar to PCA, the kernel PCA can be sensitive to outliers. Various approaches have been proposed in the literature to robustify the classical PCA. However, it is not immediately clear how these approaches can be ...