Wrinkle Reduction

An analysis of large data sets from an empirical and geometric viewpoint Data reduction is a rapidly emerging field with broad applications in essentially all fields where large data sets are collected and analyzed. Geometric Data Analysis is the first textbook to focus on the geometric approach to this problem of developing and distinguishing subspace and submanifold techniques for low-dimensional data representation. Understanding the geometrical nature of the data under investigation is presented as the key to identifying a proper reduction technique. Focusing on the construction of dimensionality-reducing mappings to reveal important geometrical structure in the data, the sequence of chapters is carefully constructed to guide the reader from the beginnings of the subject to areas of current research activity. A detailed, and essentially self-contained, presentation of the mathematical prerequisites is included to aid readers from a broad variety of backgrounds. Other topics discussed in Geometric Data Analysis include: The Karhunen-Loeve procedure for scalar and vector fields with extensions to missing data, noisy data, and data with symmetryNonlinear methods including radial basis functions (RBFs) and backpropa-gation neural networksWavelets and Fourier analysis as analytical methods for data reductionExpansive discussion of recent research including the Whitney reduction network and adaptive bases codeveloped by the authorAnd much more The methods are developed within the context of many real-world applications involving massive data sets, including those generated by digital imaging systems and computer simulations of physical phenomena. Empirically based representationsare shown to facilitate their investigation and yield insights that would otherwise elude conventional analytical tools.

The use of symmetries and conservation laws in the qualitative description of dynamics has a long history going back to the founders of classical mechanics. In some instances, the symmetries in a dynamical system can be used to simplify its kinematical description via an important procedure that has evolved over the years and is known generically as reduction. The focus of this work is a comprehensive and self-contained presentation of the intimate connection between symmetries, conservation laws, and reduction, treating the singular case in detail. The exposition reviews the necessary prerequisites, beginning with an introduction to Lie symmetries on Poisson and symplectic manifolds. This is followed by a discussion of momentum maps and the geometry of conservation laws that are used in the development of symplectic reduction.

Polynomial-time reduction - In computational complexity theory a polynomial-time reduction is a reduction which is computable by a deterministic Turing machine in polynomial time. If it is a many-one reduction, it is called a polynomial-time many-one reduction, polynomial transformation, or Karp reduction.

Truth table reduction - In computability theory truth table reduction is a reduction which is stronger than many-one reduction but weaker than Turing reduction.

Reduction (mathematics) - In linear algebra, reduction is the process of applying rules to a series of equations or matrices to change them into a "simpler" format. These rules involve manipulating either the rows or the columns of a matrix, and so this is usually referred to as row-reduction or column-reduction.

Turing reduction - In computability theory, a Turing reduction from a problem A to a problem B is, intuitively, a reduction which easily solves A, assuming B is easy to solve. More formally, a Turing reduction is a function computable by an oracle machine with an oracle for B.

wrinklereduction

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