Graph Kernels

We present a unified framework to study graph kernels, special cases of which include the random walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time complexity of kernel computation between unlabeled graphs with n vertices from O(n^6) to O(n^3). We find a spectral decomposition approach even more efficient when computing entire kernel matrices. For labeled graphs we develop conjugate gradient and fixed-point methods that take O(dn^3) time per iteration, where d is the size of the label set. By extending the necessary linear algebra to Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for d-dimensional edge kernels, and O(n^4) in the infinite-dimensional case; on sparse graphs these algorithms only take O(n^2) time per iteration in all cases. Experiments on graphs from bioinformatics and other application domains show that these techniques can speed up computation of the kernel by an order of magnitude or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to R-convolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment kernel of Fröhlich et al. (2006) yet provably positive semi-definite

Positive and generalized positive real lemma for slice hyperholomorphic functions

In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part in the half space of quaternions with positive real part, as well as the case of (generalized) Schur functions in the open unit ball

A new realization of rational functions, with applications to linear combination interpolation

We introduce the following linear combination interpolation problem (LCI): Given $N$ distinct numbers $w_1,\ldots w_N$ and $N+1$ complex numbers $a_1,\ldots, a_N$ and $c$, find all functions $f(z)$ analytic in a simply connected set (depending on $f$) containing the points $w_1,\ldots,w_N$ such that $$\sum_{u=1}^Na_uf(w_u)=c.$$ To this end we prove a representation theorem for such functions $f$ in terms of an associated polynomial $p(z)$. We first introduce the following two operations, $(i)$ substitution of $p$, and $(ii)$ multiplication by monomials $z^j, 0\le j < N$. Then let $M$ be the module generated by these two operations, acting on functions analytic near $0$. We prove that every function $f$, analytic in a neighborhood of the roots of $p$, is in $M$. In fact, this representation of $f$ is unique. To solve the above interpolation problem, we employ an adapted systems theoretic realization, as well as an associated representation of the Cuntz relations (from multi-variable operator theory.) We study these operations in reproducing kernel Hilbert space): We give necessary and sufficient condition for existence of realizations of these representation of the Cuntz relations by operators in certain reproducing kernel Hilbert spaces, and offer infinite product factorizations of the corresponding kernels

Rational inner functions in the Schur-Agler class of the polydisk

Every two variable rational inner function on the bidisk has a special representation called a transfer function realization. It is well known and related to important ideas in operator theory that this does not extend to three or more variables on the polydisk. We study the class of rational inner functions on the polydisk which do possess a transfer function realization (the Schur-Agler class) and investigate minimality in their representations. Schur-Agler class rational inner functions in three or more variables cannot be represented in a way that is as minimal as two variables might suggest.Comment: 14 page

Nevanlinna-Pick interpolation on distinguished varieties in the bidisk

This article treats Nevanlinna-Pick interpolation in the setting of a special class of algebraic curves called distinguished varieties. An interpolation theorem, along with additional operator theoretic results, is given using a family of reproducing kernels naturally associated to the variety. The examples of the Neil parabola and doubly connected domains are discussed.Comment: 31 pages. The question left open at the end of version 1 has been answered in the affirmative; see Theorem 1.12 and Corollary 1.13 in version

Schur functions and their realizations in the slice hyperholomorphic setting

we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels, positive definite functions in this setting and we show how they can be obtained in our setting using the extension operator and the slice regular product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges-Rovnyak space