Simplicial Nonlinear Principal Component Analysis

We present a new manifold learning algorithm that takes a set of data points lying on or near a lower dimensional manifold as input, possibly with noise, and outputs a simplicial complex that fits the data and the manifold. We have implemented the algorithm in the case where the input data can be triangulated. We provide triangulations of data sets that fall on the surface of a torus, sphere, swiss roll, and creased sheet embedded in a fifty dimensional space. We also discuss the theoretical justification of our algorithm.Comment: 21 pages, 6 figure

Ultrametric embedding: application to data fingerprinting and to fast data clustering

We begin with pervasive ultrametricity due to high dimensionality and/or spatial sparsity. How extent or degree of ultrametricity can be quantified leads us to the discussion of varied practical cases when ultrametricity can be partially or locally present in data. We show how the ultrametricity can be assessed in text or document collections, and in time series signals. An aspect of importance here is that to draw benefit from this perspective the data may need to be recoded. Such data recoding can also be powerful in proximity searching, as we will show, where the data is embedded globally and not locally in an ultrametric space.Comment: 14 pages, 1 figure. New content and modified title compared to the 19 May 2006 versio

Bifurcations of the Lagrangian orbits from the classical to the curved 3-body problem

We consider the 3-body problem of celestial mechanics in Euclidean, elliptic, and hyperbolic spaces, and study how the Lagrangian (equilateral) relative equilibria bifurcate when the Gaussian curvature varies. We thus prove the existence of new classes of orbits. In particular, we find some families of isosceles triangles, which occur in elliptic space.Comment: 26 pages, 3 figure

Fast, Linear Time, m-Adic Hierarchical Clustering for Search and Retrieval using the Baire Metric, with linkages to Generalized Ultrametrics, Hashing, Formal Concept Analysis, and Precision of Data Measurement

We describe many vantage points on the Baire metric and its use in clustering data, or its use in preprocessing and structuring data in order to support search and retrieval operations. In some cases, we proceed directly to clusters and do not directly determine the distances. We show how a hierarchical clustering can be read directly from one pass through the data. We offer insights also on practical implications of precision of data measurement. As a mechanism for treating multidimensional data, including very high dimensional data, we use random projections.Comment: 17 pages, 45 citations, 2 figure

Ultrametric Component Analysis with Application to Analysis of Text and of Emotion

We review the theory and practice of determining what parts of a data set are ultrametric. It is assumed that the data set, to begin with, is endowed with a metric, and we include discussion of how this can be brought about if a dissimilarity, only, holds. The basis for part of the metric-endowed data set being ultrametric is to consider triplets of the observables (vectors). We develop a novel consensus of hierarchical clusterings. We do this in order to have a framework (including visualization and supporting interpretation) for the parts of the data that are determined to be ultrametric. Furthermore a major objective is to determine locally ultrametric relationships as opposed to non-local ultrametric relationships. As part of this work, we also study a particular property of our ultrametricity coefficient, namely, it being a function of the difference of angles of the base angles of the isosceles triangle. This work is completed by a review of related work, on consensus hierarchies, and of a major new application, namely quantifying and interpreting the emotional content of narrative.Comment: 49 pages, 15 figures, 52 citation

Point vortices on the hyperbolic plane

We investigate some properties of the dynamical system of point vortices on the hyperboloid. This system has noncompact symmetry SL(2, R) and a coadjoint equivariant momentum map J. The relative equilibrium conditions are found and the trajectories of relative equilibria with non-zero momentum value are described. We also provide the classification of relative equilibria and the stability criteria for a number of cases, focusing on N=2, 3. Contrary to the system on the sphere, relative equilibria with non-compact momentum isotropy subgroup are found, and are used to illustrate the different stability types of relative equilibria.Comment: To appear in J. Mathematical Physic