Institute of Mathematics AS CR, v. v. i.
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-means: An iterative batch -means algorithm for big data clustering
Full text linksummary:Information technologies such as social media, mobile computing, and the realization of the industrial Internet of Things (IoT) produce huge amounts of data every day. The development of powerful tools for knowledge-discovery is imperative to deal with such a volume of data. Clustering methods are among the most important knowledge-discovery techniques. The growth in computational power and algorithmic developments allow us to efficiently and accurately solve clustering problems in large datasets. However, these developments are insufficient to deal with clustering problems in big datasets. This is because these datasets cannot be processed as a whole due to hardware and computational restrictions. In this paper an iterative batch -means (-means) algorithm is proposed that yields good clustering results with low computation costs on big datasets. It is designed to cluster datasets using batch data. The efficiency and accuracy of the proposed algorithm are investigated depending on the size of batches, the number of attributes and clusters. The algorithm is compared with the classic -means and mini batch -means algorithms using computational results on several real-world datasets, all of which are available from the UCI Machine Learning Repository. The smallest dataset has 500000 data points and 2 attributes and the largest one contains 43930257 data points and 16 attributes. Results demonstrated that the -means algorithm outperforms both the -means and mini batch -means algorithms in the sense of both efficiency and accuracy and it is applicable for the clustering of big datasets. The proposed algorithm provides real time clustering and may have direct applications in expert and intelligent systems. Furthermore, results from this paper will have a clear impact in the sense of designing more accurate and efficient clustering algorithms for big datasets taking into account available computer resources
Adaptive inverse optimal control for unstable reaction-diffusion PDE system
Full text linksummary:We study adaptive inverse optimal boundary control for reaction-diffusion PDE system with unknown coefficient. First, an adaptive boundary control with parameter update rule is designed which no attempt is made to force parameter convergence. Next, it is proven through a non quadratic Lyapunov function that the closed-loop system is globally asymptotically stable. Further, it indicates that adaptive boundary control with parameter update law is optimal for a meaningful functional. Finally, the effectiveness of the proposed control design is demonstrated through an example
On quasirecurrent manifolds
Full text linksummary:We introduce a type of Riemannian manifolds (namely, quasirecurrent manifold) and study its several geometric properties. Among others, we prove that the scalar curvature of such a manifold is constant, and that the manifold is Einstein under certain condition. In addition, we deal with a quasirecurrent product manifold. Finally, we ensure the existence of quasirecurrent manifold by a proper example
Shape analysis and comparison of audio patterns using divergence measures
No full textsummary:We represent the point clouds of objects and audio signals as manifolds of Gaussian Mixture Models, and analyze the shape variation and compare the audio patterns using three divergence measures, namely the Kullback-Leibler Divergence, Jensen-Shannon Divergence, and Modified Symmetric Kullback-Leibler Divergence. Experiments are conducted on basic geometric shapes, 3D human body shapes, animal shapes, point clouds of the same object produced from the dense point clouds in the PU-GAN (Point Cloud Upsampling Adversarial Network) dataset. Then, we present a method to generate a point cloud of an audio signal using the Short-Time Fourier Transform. The audio-derived point clouds represent frequency, time, and magnitude relationships, enabling analysis of speech and audio patterns. The results across all datasets show that the Modified Symmetric Kullback-Leibler Divergence provides the most distinct and stable comparison between different point clouds, demonstrating its robustness for point cloud comparison
LMI-based nonlinear observer design for a class of nonlinear systems modeled with differential algebraic equations
Full text linksummary:This work presents a novel methodology to design nonlinear observers for a class of systems modeled as differential algebraic equations. The proposal is based on writing both the system and the observer as nonlinear descriptor redundancy representations subject to algebraic restrictions; then the nonlinear observation error system is written in an explicit incremental form via suitable factorization techniques. A redundant Lyapunov function is then employed to guarantee asymptotic stability of the estimation error; linearity of the Lyapunov function and its time derivative with respect to the observer gains and Lyapunov function terms, allows gridding or convex treatment of expressions via linear matrix inequalities. Physical examples are presented to illustrate the proposal effectiveness against former methodologies
A note on -flat preenvelopes
No full textsummary:We investigate the notion of -flat preenvelopes of modules. In particular, we give an example that a ring being coherent does not imply that every -module has an -flat preenvelope, giving a negative answer to the question proposed by D. Bennis and A. Bouziri (2025). Besides, we also show that a ring being coherent also does not imply that is an -coherent ring in general
Weighted norm inequalities for certain classes of multilinear operators on Morrey spaces
No full textsummary:By applying a new class of multiple weights containing the classical multiple weights, we establish strong-type and weak-type endpoint estimates for a certain class of multilinear operators, as well as for commutators generated by the new BMO function, on weighted Morrey spaces