In this article we extend the (recently published) unsupervised information theoretic vector quantization approach based on the Cauchy–Schwarz-divergence for matching data and prototype densities to supervised learning and classification. In particular, first we generalize the unsupervised method to more general metrics instead of the Euclidean, as it was used in the original algorithm. Thereaf...

The constant γ in the strengthened Cauchy-Bunyakowskii-Schwarz inequality is a basic tool for the construction of two-level and multilevel preconditioning matrices. Therefore many authors consider estimates or computations of this quantity. In this paper the bilinear form arising from 3D linear elasticity problems is considered on a polyhedron. The cosine of the abstract angle between multileve...

We developed parallel time domain decomposition methods to solve systems of linear ordinary differential equations (ODEs) based on the Aitken-Schwarz [5] or primal Schur complement domain decomposition methods [4]. The methods require the transformation of the initial value problem in time defined on ]0,T ] into a time boundary values problem. Let f (t,y(t)) be a function belonging to C 1(R+,Rd...

The time dependence of nonclassical correlations is investigated for two fields (1,2) generated by an ensemble of cold cesium atoms via the protocol of Duan et al. [Nature (London) 414, 413 (2001)]]. The correlation function R(t1,t2) for the ratio of cross to autocorrelations for the (1,2) fields at times (t1,t2) is found to have a maximum value R(max=292+/-57, which significantly violates the ...

H ·H := ̆ h = fg : f, g ∈ H ̄ = H ←↩ H is the product space of H2, by inner/outer factorization and Cauchy-Schwarz inequality. It is interesting, then, to find the dual space of H1. C. Fefferman [7] proved that, under the H2 paring (with some care), (H2 ·H2)∗ = (H1)∗ = BMO∩H(D) is the space of the analytic functions with bounded mean oscillation. The definition of BMO, born out of a problem in el...

It is well known that a continuous linear functional is bounded, and finding the bound or norm of a continuous linear functional is a fundamental task in functional analysis. Recently, in light of the Taylor formula and the Cauchy-Schwarz inequality, Gavrea and Ivan in 1 obtained an inequality for the continuous linear functionalL satisfying 1.1 . In order to state their result, we need some mo...

This paper contributes a tutorial level discussion of some interesting properties of the recent Cauchy–Schwarz (CS) divergence measure between probability density functions. This measure brings together elements from several different machine learning fields, namely information theory, graph theory and Mercer kernel and spectral theory. These connections are revealed when estimating the CS dive...

In the Nordic countries, with their longstanding tradition of valuing education for masses, a large number schools have been built over more than century-long period and therefore represen...