Jacek Tabor

Cross-entropy clustering

We build a general and easily applicable clustering theory, which we call crossentropy clustering (shortly CEC), which joins the advantages of classical kmeans (easy implementation and speed) with those of EM (ane invariance and ability to adapt to clusters of desired shapes). Moreover, contrary to k-means and EM, CEC nds the optimal number of clusters by automatically removing groups which have negative information cost.

On a linear iterative equation

We consider the following iterative equation % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Two ellipsoid Support Vector Machines

\sum_{i=0}^{k}a_{i}f^{i}(x)=0,

Applications of de Rham theorem in approximate midconvexity

where a0,…, ak are given real numbers and ƒ is an unknown function. Assuming some conditions on the coefficients a0,…, ak we prove that this equation has exactly one solution and that the solution depends continuously on the coefficients.

Detection of Elliptical Shapes via Cross-Entropy Clustering

We propose a new entropy based multithreshold linear classifier with an adaptive kernel density estimation.Proposed classifier maximizes multiple margins, while being conceptually similar in nature to SVM.This method gives good classification results and is especially designed for unbalanced datasets.It achieves significantly better results than SVM as part of an expert system designed for drug discovery.Resulting model provides insight into the internal data geometry and can detect multiple clusters. This paper proposes a new multithreshold linear classifier (MELC) based on the Renyis quadratic entropy and Cauchy-Schwarz divergence, combined with the adaptive kernel density estimation in the one dimensional projections space. Due to its nature MELC is especially well adapted to deal with unbalanced data. As the consequence of both used model and the applied density regularization technique, it shows strong regularization properties and therefore is almost unable to overfit. Moreover, contrary to SVM, in its basic form it has no free parameters, however, at the cost of being a non-convex optimization problem which results in the existence of local optima and the possible need for multiple initializations.In practice, MELC obtained similar or higher scores than the ones given by SVM on both synthetic and real data from the UCI repository. We also perform experimental evaluation of proposed method as a part of expert system designed for drug discovery problem. It appears that not only MELC achieves better results than SVM but also gives some additional insights into data structure, resulting in more complex decision support system.

Entropy of the Mixture of Sources and Entropy Dimension

Abstract In classification problems classes usually have different geometrical structure and therefore it seems natural for each class to have its own margin type. Existing methods using this principle lead to the construction of the different (from SVM) optimization problems. Although they outperform the standard model, they also prevent the utilization of existing SVM libraries. We propose an approach, named 2 eSVM , which allows use of such method within the classical SVM framework. This enables to perform a detailed comparison with the standard SVM. It occurs that classes in the resulting feature space are geometrically easier to separate and the trained model has better generalization properties. Moreover, based on evaluation on standard datasets, 2 eSVM brings considerable profit for the linear classification process in terms of training time and quality. We also construct the 2 eSVM kernelization and perform the evaluation on the 5-HT2A ligand activity prediction problem (real, fingerprint based data from the cheminformatic domain) which shows increased classification quality, reduced training time as well as resulting model’s complexity.

Stability of the generalized alternative cauchy equation

Let G be an amenable metric semigroup with nonempty center, let E be a reflexive Banach space, and let ƒ: G → E be a given function. By Cƒ: G × G → E we understand the Cauchy difference of the function /, i.e.: % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!