Jacob Kogan

Grouping Multidimensional Data

Clustering is one of the most fundamental and essential data analysis techniques. Clustering can be used as an independent data mining task to discern intrinsic characteristics of data, or as a preprocessing step with the clustering results then used for classification, correlation analysis, or anomaly detection. Kogan and his co-editors have put together recent advances in clustering large and high-dimension data. Their volume addresses new topics and methods which are central to modern data analysis, with particular emphasis on linear algebra tools, opimization methods and statistical techniques. The contributions, written by leading researchers from both academia and industry, cover theoretical basics as well as application and evaluation of algorithms, and thus provide an excellent state-of-the-art overview. The level of detail, the breadth of coverage, and the comprehensive bibliography make this book a perfect fit for researchers and graduate students in data mining and in many other important related application areas.

Feature Selection and Document Clustering

Feature selection is a basic step in the construction of a vector space or bag-of-words model [BB99]. In particular, when the processing task is to partition a given document collection into clusters of similar documents a choice of good features along with good clustering algorithms is of paramount importance. This chapter suggests two techniques for feature or term selection along with a number of clustering strategies. The selection techniques significantly reduce the dimension of the vector space model. Examples that illustrate the effectiveness of the proposed algorithms are provided.

Frequency domain criterion for robust stability of interval time-delay systems

Abstract In this paper we characterize the boundary ∂f ( B ) of the image f ( B ) of a box B in R m under a nonlinear mapping f : R m → C . We generalize results recently reported by Polyak and Kogan (1993) [Necessary and Sufficient Conditions for Robust Stability of Multiaffine Systems. Mathematics Research Report 93-06, University of Maryland Baltimore County] for multiaffine mappings, and provide computationally tractable necessary and sufficient robust stability conditions for quasipolynomials with interval coefficients and interval delays. A numerical stability verification for a quasipolynomial family with two interval delays is presented.

Computation of stability radius for families of bivariate polynomials

In this paper we apply optimization techniques to the problem of robust stability of a family of bivariate polynomials under affine coefficient perturbations. The size of the perturbations is measured by a convex function. In this paper we concentrate onlp weighted norms, for three special casesp = 1,p = 2, andp = ∞. Necessary and sufficient conditions for robust stability are provided. Evaluation of the stability radius is reduced to a minimization problem in two-dimensional space. The results open the door to the development and implementation of reliable and efficient algorithms for the computation of the stability radius.

Data Driven Similarity Measures for k-Means Like Clustering Algorithms

We present an optimization approach that generates k-means like clustering algorithms. The batch k-means and the incremental k-means are two well known versions of the classical k-means clustering algorithm (Duda et al. 2000). To benefit from the speed of the batch version and the accuracy of the incremental version we combine the two in a “ping–pong” fashion. We use a distance-like function that combines the squared Euclidean distance with relative entropy. In the extreme cases our algorithm recovers the classical k-means clustering algorithm and generalizes the Divisive Information Theoretic clustering algorithm recently reported independently by Berkhin and Becher (2002) and Dhillon1 et al. (2002). Results of numerical experiments that demonstrate the viability of our approach are reported.

On exponential stability of linear systems and Hurwitz stability of characteristic quasipolynomials

Abstract A family of linear systems with uncertainties in parameters and delays, and its corresponding family of characteristic quasipolynomials, are considered. Both commensurate and non-commensurate delays are considered. It is shown that for a large class of systems, namely those which comply with a simple condition ensuring no degree reduction, Hurwitz stability implies also exponential stability.

Exponential stability of linear systems with commensurate time-delays

In this paper we consider a family of linear systems with commensurate time-delays. Each coefficient of the characteristic quasipolynomial and the delay is allowed to vary in prespecified intervals. The main result of the paper is a necessary and sufficient condition for stability of the family of quasipolynomials. This yields a computationally tractable numerical procedure which determines whether or not the family is exponentially stable.

Applications of optimization methods to robust stability of linear systems

We study the robust stability problem for a family of polynomials. We allow for all the coefficients of the polynomials to be affinely perturbed, where the size of the perturbation is measured by an arbitrary convex function. We apply optimization techniques, and in particular convex duality methods, to derive simple formulas for the stability radius, to find a minimal perturbation which destroys stability, and to obtain necessary and sufficient conditions for robust stability. Our framework is general enough to cover many applications. As special cases, we obtain many results recently reported in the literature.

Building initial partitions through sampling techniques

A variety of iterative clustering algorithms require an initial partition of a dataset as an input parameter. As a rule a good choice of the initial partition is essential for building a high quality final partition. In this note, we generate initial partitions by using small samples of the data. Numerical experiments with k-means like clustering algorithms are reported.

On stability of quasipolynomials with annular uncertainties and interval commensurate delays

The paper concerns the Hurwitz stability of a family of quasipolynomials with commensurate delays. Each coefficient of the quasipolynomials belongs to a prescribed annulus in the complex plane, and the delay belongs to a prescribed real interval. A computationally tractable robust stability criterion is the main result of the paper.