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The emerging field of computational topology utilizes theory from topology and the power of computing to solve problems in diverse fields. Recent applications include computer graphics, computer-aided design (CAD), and structural biology, all of which involve understanding the intrinsic shape of some real or abstract space. A primary goal of this book is to present basic concepts from topology and Morse theory to enable a non-specialist to grasp and participate in current research in computational topology. The author gives a self-contained presentation of the mathematical concepts from a computer scientist's point of view, combining point set topology, algebraic topology, group theory, differential manifolds, and Morse theory. He also presents some recent advances in the area, including topological persistence and hierarchical Morse complexes. Throughout, the focus is on computational challenges and on presenting algorithms and data structures when appropriate.
Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical phenomena. The present book introduces the reader to the general principles underlying these methods and then describes in detail their convergence properties when applied to ordinary differential equations, functional equations with (Volterra type) memory terms, delay equations, and differential-algebraic and integral-algebraic equations. Each chapter starts with a self-contained introduction to the relevant theory of the class of equations under consideration. Numerous exercises and examples are supplied, along with extensive historical and bibliographical notes utilising the vast annotated reference list of over 1300 items. In sum, Hermann Brunner has written a treatise that can serve as an introduction for students, a guide for users, and a comprehensive resource for experts.
In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing, it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a powerful tool which work well in very general circumstances and so are becoming of widespread use as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent. The author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.
In this book, we study a system which perceives the real world. Such a system has to estimate an information source by observation. If the information source is a probability distribution, then the estimation process is called statistical learning, and the system is said to be a statistical model or a learning machine.
A lot of statistical models have hierarchical layers, hidden variables, a collection of modules, or grammatical structures. Such models are nonidentifiable and contain singularities in their parameter spaces. In fact, the map from a parameter to a statistical model is not one-to-one, and the Fisher information matrix is not positive definite. Such statistical models are called singular. It has been difficult to examine the learning process of singular models, because there has been no mathematical theory for such models.
In this book, we establish a mathematical foundation which enables us to understand the learning process of singular models. This chapter gives an overview of the book before a rigorous mathematical foundation is developed.
Basic concepts in statistical learning
To describe what statistical learning is, we need some basic concepts in probability theory. For the reader who is unfamiliar with probability theory, Section 1.6 summarizes the key results.
Random samples
Let N be a natural number and ℝN be the N-dimensional real Euclidean space. We study a case when information data are represented by vectors in ℝN.
In this book, we introduce a fundamental relation between algebraic geometry and statistical learning theory.
A lot of statistical models and learning machines used in information science, for example, mixtures of probability distributions, neural networks, hidden Markov models, Bayesian networks, stochastic context-free grammars, and topological data analysis, are not regular but singular, because they are nonidentifiable and their Fisher information matrices are singular. In such models, knowledge to be discovered from examples corresponds to a singularity, hence it has been difficult to develop a mathematical method that enables us to understand statistical estimation and learning processes.
Recently, we established singular learning theory, in which four general formulas are proved for singular statistical models. Firstly, the log likelihood ratio function of any singular model can be represented by the common standard form even if it contains singularities. Secondly, the asymptotic behavior of the evidence or stochastic complexity is clarified, giving the result that the learning coefficient is equal to the maximum pole of the zeta function of a statistical model. Thirdly, there exist equations of states that express the universal relation of the Bayes quartet. We can predict Bayes and Gibbs generalization errors using Bayes and Gibbs training errors without any knowledge of the true distribution. And lastly, the symmetry of the generalization and training errors holds in the maximum likelihood and a posteriori estimators. If one-point estimation is applied to statistical learning, the generalization error is equal to the maximum value of a Gaussian process on a real analytic set.
In this chapter, we study statistical model evaluation and statistical hypothesis tests in singular learning machines. Firstly, we show that there is no universally optimal learning in general and that model evaluation and hypothesis tests are necessary in statistics. Secondly, we analyze two information criteria: stochastic complexity and generalization error in singular learning machines. Thirdly, we show a method to produce a statistical hypothesis test if the null hypothesis is a singularity of the alternative hypothesis. Then the methods by which the Bayes a posteriori distribution is generated are introduced. We discuss the Markov chain Monte Carlo and variational approximation. In the last part of this chapter, we compare regular and singular learning theories. Regular learning theory is based on the quadratic approximation of the log likelihood ratio function and the central limit theorem on the parameter space, whereas singular learning theory is based on the resolution of singularities and the central limit theorem on the functional space. Mathematically speaking, this book generalizes regular learning theory to singular statistical models.
Universally optimal learning
There are a lot of statistical estimation methods. One might expect that there is a universally optimal method, which always gives a smaller generalization error than any other method. However, in general, such a method does not exist.
Assumption. Assume that Φ(w) is the probability density function on ℝd, and that a parameter ω is chosen with respect to Φ(ω).