This paper derives three competitive learning mechanisms from first principles to obtain clusters of comparable sizes when both inputs and representatives are normalized. These mechanisms are very effective in achieving balanced grouping of inputs in high dimensional spaces, as illustrated by experimental results on clustering two popular text data sets in 26,099 and 21,839 dimensional spaces r...

The topic of this lecture is dimensionality reduction. Many problems have been efficiently solved in low dimensions, but very often the solution to low-dimensional spaces are impractical for high dimensional spaces because either space or running time is exponential in dimension. In order to address the curse of dimensionality, one technique is to map a set of points in a high dimensional space...

In this paper, we deal with the subdierential concept onHadamard spaces. Flat Hadamard spaces are characterized, and nec-essary and sucient conditions are presented to prove that the subdif-ferential set in Hadamard spaces is nonempty. Proximal subdierentialin Hadamard spaces is addressed and some basic properties are high-lighted. Finally, a density theorem for subdierential set is established.

Biological processes such as contagious disease spread patterns and tumor growth dynamics are modelled using a set of coupled differential equations. Experimental data is usually used to calibrate models so they can be used to make future predictions. In this study, numerical methods were implemented to approximate solutions to mathematical models that were not solvable analytically, such as a ...

Another facet of geodesics is GAT or “geometric approximation theory”. It is very difficult to work directly with infinite dimensional geometry and GAT is a special method in projective K’́ahler geometry to make fiinite dimensional approximations of the infinite dimensional locally symmetric space Hω by genuine finite dimensional symmetric spaces of type GC/G with G = SU(N). The compact group SU...

We study approximation of embeddings between finite dimensional Lp spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The results show that for certain regions of the parameter domain quantum computation can essentially improve the rate of convergence of classical deterministic or...

Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky–Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodins...