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Analogy as Possible Identity within Pure Inductive Logic
Get PDFWithin the framework of Pure Inductive Logic we investigate a regularity principle of Analogy based on the interpretation of analogy as possible identity and locate it with respect to the central principles of Constant and Spectrum Exchangeability
Conversions between barycentric, RKFUN, and Newton representations of rational interpolants
No full textWe derive explicit formulas for converting between rational interpolants in barycentric, rational Krylov (RKFUN), and Newton form. We show applications of these conversions when working with rational approximants produced by the AAA algorithm [Y. Nakatsukasa, O. Sete, L. N. Trefethen, SIAM J. Sci. Comput. 40(3), 2018] within the Rational Krylov Toolbox and for the solution of nonlinear eigenvalue problems
Computational Information Geometry For Image and Signal Processing
Get PDFThis book focuses on the application and development of information geometric methods in the analysis, classification and retrieval of images and signals. It provides introductory chapters to help those new to information geometry and applies the theory to several applications. This area has developed rapidly over recent years, propelled by the major theoretical developments in information geometry, efficient data and image acquisition and the desire to process and interpret large databases of digital information. The book addresses both the transfer of methodology to practitioners involved in database analysis and in its efficient computational implementation
Dimensionality Reduction for Information Geometric Characterization of Surface Topographies
Get PDFStochastic textures with features spanning many length scales arise in a range
of contexts in physical and natural sciences, from nanostructures like synthetic bone
to ocean wave height distributions and
cosmic phenomena like inter-galactic cluster void distributions.
Here we used a data set of 35 surface topographies, each of 2400x2400 pixels
with spatial resolution between 4~\mum\ and 7~\mum\ per pixel, and fitted trivariate
Gaussian distributions to represent their spatial structures.
For these we computed pairwise information metric distances using the Fisher-Rao metric.
Then dimensionality reduction was used to reveal the groupings among subsets of samples
in an easily comprehended graphic in 3-space. The samples here came from the
papermaking industry but such a reduction of large frequently noisy spatial data
sets is useful in a range of materials and contexts at all scales
The Finite Values Property
Get PDFWe argue that the simplicity condition on a probability function on sentences of a predicate language that it takes only finitely many values on the sentences of any finite sublanguage of can be viewed as rational. We then go on to investigate consequences of this condition, linking it to the model theoretic notion of quantifier elimination
Collocation methods for exploring perturbations in linear stability analysis
Get PDFEigenvalue analysis is a well-established tool for stability analysis of dynamical systems. However, there are situations where eigenvalues miss some important features of physical models. For example, in models of incompressible fluid dynamics, there are examples where linear stability analysis predicts stability but transient simulations exhibit significant growth of infinitesimal perturbations. This behavior can be predicted by pseudo-spectral analysis. In this study, we show that an approach similar to pseudo-spectral analysis can be performed inexpensively using stochastic collocation methods and the results can be used to provide quantitative information about instability. In addition, we demonstrate that the results of the perturbation analysis provide insight into the behavior of unsteady flow simulations
The Nonlinear Eigenvalue Problem
Get PDFNonlinear eigenvalue problems arise in a variety of science and engineering applications and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton's method, contour integration, and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods
Computing the Action of Trigonometric and Hyperbolic Matrix Functions
Get PDFWe derive a new algorithm for computing the action of the cosine,
sine, hyperbolic cosine, and
hyperbolic sine of a matrix on a matrix ,
without first computing .
The algorithm can compute
and simultaneously, and likewise for
and ,
and it uses only real arithmetic when is real.
The algorithm exploits an existing algorithm \texttt{expmv}
of Al-Mohy and Higham for
and its underlying backward error analysis.
Our experiments show that the new algorithm performs in
a forward stable manner and is generally significantly faster than
alternatives based on multiple invocations of \texttt{expmv}
through formulas such as
Translation Invariance and Miller's Weather Example
Get PDFIn his 1974 paper "Popper's Qualitative Theory of Verisimilitude" published in the British Journal for the Philosophy of Science David Miller gave his so called `Weather Example' to argue that the Hamming distance between constituents is flawed as a measure of proximity
to truth since the former is not, unlike the latter, translation invariant. In this present paper we generalise David Miller's Weather Example in both the unary and polyadic cases, characterising precisely which permutations of constituents/atoms can be effected by translations.
In turn this suggests a meta-principle of the rational assignment of subjective probabilities, that rational principles should be preserved under translations, which we formalise and give a particular characterisation of in the context of Unary Pure Inductive Logic
The Structured Condition Number of a Differentiable Map Between Matrix Manifolds, with Applications
Get PDFWe study the structured condition number of differentiable maps between smooth matrix manifolds, developing a theoretical framework that extends previous results for vector subspaces to any smooth manifold. We present algorithms to compute the structured condition number. As special cases of smooth manifolds, we analyze automorphism groups, and Lie and Jordan algebras associated with a scalar product. For such manifolds, we derive a lower bound on the structured condition number that is cheaper to compute than the structured condition number. We provide numerical comparisons between the structured and unstructured condition numbers for the principal matrix logarithm and principal matrix square root of matrices in automorphism groups as well as for the map between matrices in automorphism groups and their polar decomposition. We show that our lower bound can be used as a good estimate for the structured condition number when the matrix argument is well conditioned. We show that the structured and unstructured condition numbers can differ by many orders of magnitude, thus motivating the development of algorithms preserving structure