Entropy Dimension Reduction Method for Randomized Machine Learning Problems
The direct and inverse projections (DIP) method was proposed to reduce the feature space to the given dimensions oriented to the problems of randomized machine learning and based on the procedure of “direct” and “inverse” design. The “projector” matrices are determined by maximizing the relative entropy. It is suggested to estimate the information losses by the absolute error calculated with the use of the Kullback–Leibler function (SRC method). An example illustrating these methods was given.
Properties of Erdos measure and the invariant Erdos measure for the golden ratio and all values of the Bernoulli parameter are studies. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdos measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet. An effective algorithm for calculating the entropy of an invariant Erdos measure is proposed. It is shown that, for certain values of the Bernulli parameter, the algorithm gives the Hausdorff dimension of an Erdos measure to 15 decimal places.
Properties of Erdos measure and the invariant Erdos measure for the golden ratio and all values of the Bernoulli parameter are studies. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdos measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet.
This volume contains the papers presented at the 6th International Conference on Similarity Search and Applications (SISAP 2013), held at A Coruna, Spain, during October 2–4, 2013. The International Conference on Similarity Search and Applications (SISAP) is an annual forum for researchers and application developers in the area of similarity data management. It aims at the technological problems shared by many application domains, such as data mining, information retrieval, computer vision, pattern recognition, computational biology, geography, biometrics, machine learning, and many others that need similarity searching as a necessary supporting service. Traditionally, SISAP conferences have put emphasis on the distance-based searching, but in general the conference concerns both the effectiveness and efficiency aspects of any similarity search approach.
In this paper, we present a modification of dynamic programming algorithms (DPA), which we denote as graphical algorithms (GrA). For some single machine scheduling problems, it is shown that the time complexity of the GrA is less than the time complexity of the standard DPA. Moreover, the average running time of the GrA is often essentially smaller. A GrA can also solve large-scale instances and instances, where the parameters are not integer. For some problems, GrA has a polynomial time complexity in contrast to a pseudo-polynomial complexity of a DPA.
Information systems have been developed in parallel with computer science, although information systems have roots in different disciplines including mathematics, engineering, and cybernetics. Research in information systems is by nature very interdisciplinary. As it is evidenced by the chapters in this book, dynamics of information systems has several diverse applications. The book presents the state-of-the-art work on theory and practice relevant to the dynamics of information systems. First, the book covers algorithmic approaches to numerical computations with infinite and infinitesimal numbers. Also the book presents important problems arising in service-oriented systems, such as dynamic composition, analysis of modern service-oriented information systems, and estimation of customer service times on a rail network from GPS data. After that, the book addresses the complexity of the problems arising in stochastic and distributed systems. In addition, the book discusses modulating communication for improving multi-agent learning convergence. Network issues, in particular minimum risk maximum clique problems, vulnerability of sensor networks, influence diffusion, community detection, and link prediction in social network analysis, as well as a comparative analysis of algorithms for transmission network expansion planning are described in subsequent chapters. We thank all the authors and anonymous referees for their advice and expertise in providing valuable contributions, which improved the quality of this book. Furthermore, we want to thank Springer for helping us to produce this book.
We revisit the problems of computing the maximal and the minimal non-empty suffixes of a substring of a longer text of length n, introduced by Babenko, Kolesnichenko and Starikovskaya [CPM’13]. For the minimal suffix problem we show that for any 1 ≤ τ ≤ logn there exists a linear-space data structure with(τ)query time and(nlogn/τ)preprocessing time. As a sample application, we show that this data structure can be used to compute the Lyndon decomposition of any substring of the text in(kτ)time, where k is the number of distinct factors in the decomposition. For the maximal suffix problem we give a linear-space structure with(1)query time and(n)preprocessing time, i.e., we manage to achieve both the optimal query and the optimal construction time simultaneously.
The formula for calculating the entropy and the Hausdorff dimension of an invariant Erdos measure for the pseudogolden ratio and all values Bernoulli parameter is obtained. This formula make possible calculating the entropy and the Hausdorff dimension with high accuracy.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.