Related papers: Lectures on piecewise distance preserving maps
We present applications of tropical geometry to some integrable piecewise-linear maps, based on the lecture given by one of the authors (R. I.) at the workshop "Tropical Geometry and Integrable Systems" (University of Glasgow, July 2011),…
I give a pedagogical introduction to the concepts and the tools that are necessary to study particle physics models in higher dimensions. I then give a more detailed presentation of warped compactifications and discuss their possible…
These are lecture notes on cut-and-paste methods in 3-dimensional contact geometry.
Many real-world machine learning problems involve inherently hierarchical data, yet traditional approaches rely on Euclidean metrics that fail to capture the discrete, branching nature of hierarchical relationships. We present a theoretical…
Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the…
Piecewise smooth systems are intensively studied today in many application areas, such as economics, finance, engineering, biology, and ecology. In this work, we consider a class of one-dimensional piecewise linear discontinuous maps with a…
Consider the map $S$ which sends a planar polygon $P$ to a new polygon $S(P)$ whose vertices are the intersection points of second nearest sides of $P$. This map is the inverse of the famous pentagram map. In this paper we investigate the…
We present in this paper a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces,…
In this paper we report some important results that help in analizing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the…
Starting with a similarity function between objects, it is possible to define a distance metric on pairs of objects, and more generally on probability distributions over them. These distance metrics have a deep basis in functional analysis,…
We study periodic, piecewise linear maps on the plane starting with the Mort Brown's map. We show that if the number of pieces is two, there is only a short list of possible periods (this fact can be seen as the crystallographic restriction…
Given a piecewise linear (PL) function $p$ defined on an open subset of $\R^n$, one may construct by elementary means a unique polyhedron with multiplicities $\D(p)$ in the cotangent bundle $\R^n\times \R^{n*}$ representing the graph of the…
This paper studies iterative schemes for measure transfer and approximation problems, which are defined through a slicing-and-matching procedure. Similar to the sliced Wasserstein distance, these schemes benefit from the availability of…
Proximity maps and regions are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in…
This paper presents a data structure that summarizes distances between configurations across a robot configuration space, using a binary space partition whose cells contain parameters used for a locally linear approximation of the distance…
This is the first report on Working Paper WP-RFM-14-01. The potential and capability of sparse representations is well-known. However, their (multivariate variable) vectorial form, which is completely fine in many fields and disciplines,…
In this paper, we overview one promising avenue of progress at the mathematical foundation of deep learning: the connection between deep networks and function approximation by affine splines (continuous piecewise linear functions in…
Using quaternions and octonions, we construct some maps from the Grassmannian of 2-dimensional planes of $\mathbb{R}^n$, $\mathrm{Gr}_2(\mathbb{R}^n)$, to the projective space $\mathbb{R}\mathrm{P}^k$, for certain values of $n$ and $k$. All…
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…
A general piecewise (including pointwise) probability distribution with space-saving notation and its hierarchical particular cases are considered. The explicit closed-form normalization, expectation, and variance formulas along with the…