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"Euclidean geometry"
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In the search for beauty : unravelling non-Euclidean geometry
Chronicles the historical attempts to prove the fifth postulate of Euclid on parallel lines that led eventually to the creation of non-Euclidean geometry.
Book
PUMA: Deep Metric Imitation Learning for Stable Motion Primitives
Imitation learning (IL) facilitates intuitive robotic programming. However, ensuring the reliability of learned behaviors remains a challenge. In the context of reaching motions, a robot should consistently reach its goal, regardless of its initial conditions. To meet this requirement, IL methods often employ specialized function approximators that guarantee this property by construction. Although effective, these approaches come with some limitations: 1) they are typically restricted in the range of motions they can model, resulting in suboptimal IL capabilities, and 2) they require explicit extensions to account for the geometry of motions that consider orientations. To address these challenges, we introduce a novel stability loss function that does not constrain the function approximator's architecture and enables learning policies that yield accurate results. Furthermore, it is not restricted to a specific state space geometry; therefore, it can easily incorporate the geometry of the robot's state space. Proof of the stability properties induced by this loss is provided and the method is empirically validated in various settings. These settings include Euclidean and non‐Euclidean state spaces, as well as first‐order and second‐order motions, both in simulation and with real robots. More details about the experimental results can be found at https://youtu.be/ZWKLGntCI6w. Imitation learning enables intuitive robotic programming but faces challenges in providing guarantees due to its data‐driven nature. To address this, existing methods constrain function approximators or assume Euclidean state spaces. Hence, a novel DNN loss function is introduced, allowing for asymptotically stable and accurate motions in the non‐Euclidean end‐effector space of manipulators, modeled as first‐ and second‐order dynamical systems.
Journal Article
Hedgehogs and their Widths in Elliptic and Hyperbolic Planes
In this paper the notion of a hedgehog in non-Euclidean geometry is introduced. More specifically, the case of 2-dimensional spaces of constant curvature is considered and, in particular, spherical hedgehogs are defined in elliptic geometry and two different notions of hedgehogs in hyperbolic geometry are studied, namely,
g
-hedgehogs and
h
-hedgehogs, which extend, respectively, the corresponding well-known notions of
g
-convex and
h
-convex curves. The construction of these curves by support functions is discussed and some relations and properties are derived. Moreover, an extended notion of a width is provided for non-Euclidean hedgehogs.
Journal Article
Understanding Higher-Order Interactions in Information Space
Methods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback–Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics.
Journal Article
A new geometry-aware non-euclidean distance metric
Many machine learning algorithms use Euclidean distance as a common metric to calculate similarities between data. However, Euclidean distance is not valid when data lie on a manifold with non-zero curvatures. Therefore, we propose a new non-parametric approach that uses curvatures to calculate distances. Curvature is an appealing feature for this purpose since it is not altered by isometries. In this paper, we propose two formulas for measuring distances on a manifold with constant curvature, and their validities are proven using the theorems of differential geometry. Utilizing these formulas, an algorithm is developed to measure the distance between a point and the center of a class. In the proposed algorithm geodesies are divided into equal linear segments, assuming that the curvature remains constant within each segment. This assumption is shown to be valid in many data spaces experimentally. Observed data near each segment are used to estimate curvatures and calculate distances within each segment. Finally, the total distance is computed by summing up the non-Euclidean lengths of all segments. The proposed method is a supervised version of
k
-means, named non-Euclidean centers. The correctness of the proposed method is validated using the Riemann tensor and its related theorems in differential geometry. Furthermore, experimental results show that our method performs well in real-world data classification applications. The space of symmetric positive definite matrices, which is often endowed with non-Euclidean metrics that induce some curvature, is used for input data representations.
Journal Article
Multiplicative Euclidean and Non-Euclidean Geometry
Differential and integral calculus, the most applicable mathematical theory, was created independently by Isaac Newton and Gottfried Wilhelm Leibnitz in the second half of the 17th century. Later, Leonard Euler redirected calculus by giving a central place to the concept of function, and thus founded analysis. Two operations, differentiation and integration, are basic in calculus and analysis. In fact, they are the infinitesimal versions of the subtraction and addition operations on numbers, respectively. From 1967 until 1970, Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. Multiplicative calculus can especially be useful as a mathematical tool for economics and finance.This book is devoted to multiplicative Euclidean and non-Euclidean geometry, summarizing the most recent contributions in this area. It will appeal to a wide audience of specialists such as mathematicians, physicists, engineers and biologists, and can be used as a textbook at the graduate level or as a reference book for several disciplines.
eBook
Projected Gradient Descent Method for Tropical Principal Component Analysis over Tree Space
Tropical Principal Component Analysis (PCA) is an analogue of the classical PCA in the setting of tropical geometry, and applied it to visualize a set of gene trees over a space of phylogenetic trees, which is a union of lower-dimensional polyhedral cones in an Euclidean space with dimension m(m−1)/2, where m is the number of leaves. In this paper, we introduce a projected gradient descent method to estimate the tropical principal polytope over the space of phylogenetic trees, and we apply it to an Apicomplexa dataset. With computational experiments against Markov Chain Monte Carlo (MCMC) samplers, we show that our projected gradient descent method yields a lower sum of tropical distances between observations and their projections onto the estimated best-fit tropical polytope, compared with the MCMC-based approach.
Journal Article
Affine Geometry and Relativity
We present the basic concepts of space and time, the Galilean and pseudo-Euclidean geometry. We use an elementary geometric framework of affine spaces and groups of affine transformations to illustrate the natural relationship between classical mechanics and theory of relativity, which is quite often hidden, despite its fundamental importance. We have emphasized a passage from the group of Galilean motions to the group of Poincaré transformations of a plane. In particular, a 1-parametric family of natural deformations of the Poincaré group is described. We also visualized the underlying groups of Galilean, Euclidean, and pseudo-Euclidean rotations within the special linear group.
Journal Article
EUCLIDEAN SURFACES WITH CONFORMAL SECOND FUNDAMENTAL FORM
In the present study, we consider surfaces in E2+d with conformal second fundamental form. We give a result for Chen surfaces to have a conformal second fundamental form. We show that every Chen surface is h-conformal. Furthermore, we give necessary and sufficient conditions for general rotational surfaces in Euclidean 4-space to become h-conformal. Finally, we prove that the Vranceanu surface is h-conformal if and only if it is either flat or minimal.
Journal Article