Related papers: Lectures on piecewise distance preserving maps
Representation learning is a fundamental task in machine learning, aiming at uncovering structures from data to facilitate subsequent tasks. However, what is a good representation for planning and reasoning in a stochastic world remains an…
These are lecture notes for a 4h mini-course held in Toulouse, May 9-12th, at the thematic school on "Quantum topology and geometry". The goal of these lectures is to (a) explain some incarnations, in the last ten years, of the idea of…
Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance,…
Unsupervised deep metric learning (UDML) focuses on learning a semantic representation space using only unlabeled data. This challenging problem requires accurately estimating the similarity between data points, which is used to supervise a…
These lectures on supersymmetry and extra dimensions are aimed at finishing undergraduate and beginning postgraduate students with a background in quantum field theory and group theory. Basic knowledge in general relativity might be…
Persistence diagrams are common descriptors of the topological structure of data appearing in various classification and regression tasks. They can be generalized to Radon measures supported on the birth-death plane and endowed with an…
Simplicial complexes form an important class of topological spaces that are frequently used in many application areas such as computer-aided design, computer graphics, and simulation. Representation learning on graphs, which are just 1-d…
The kinematics of particles and rigid bodies in the plane are investigated up to higher-order accelerations. Discussion of point trajectories leads from higher-order poles to higher-order Bresse circles of the moving plane. Symplectic…
This is a pedagogical introduction covering maps of metric spaces, Gromov-Hausdorff distance and its "physical" meaning, and dilation structures as a convenient simplification of an exhaustive database of maps of a metric space into…
The field of partial differential equations (PDEs) is vast in size and diversity. The basic reason for this is that essentially all fundamental laws of physics are formulated in terms of PDEs. In addition, approximations to these…
In this paper, we introduce a generalized piecewise translation map on the Euclidean space. We provide a special case when this map is always of finite type. For a finite type map in this case, we form conjectures on the semi-continuity of…
Recent work has explored methods for learning continuous vector space word representations reflecting the underlying semantics of words. Simple vector space arithmetic using cosine distances has been shown to capture certain types of…
Manifold learning is a hot research topic in the field of computer science and has many applications in the real world. A main drawback of manifold learning methods is, however, that there is no explicit mappings from the input data…
Persistence diagram (PD) bundles, a generalization of vineyards, were introduced as a way to study the persistent homology of a set of filtrations parameterized by a topological space $B$. In this paper, we present an algorithm for…
A flat membrane with given shape is displayed; two points in the membrane are randomly selected; the probability that the separation between the points have a specified value is sought. A simple method to evaluate the probability density is…
Laplacian-based methods are popular for the dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance…
Distance Geometry is based on the inverse problem that asks to find the positions of points, in a Euclidean space of given dimension, that are compatible with a given set of distances. We briefly introduce the field, and discuss some open…
Piecewise Linear Interface Construction (PLIC) is frequently used to geometrically reconstruct fluid interfaces in Computational Fluid Dynamics (CFD) modeling of two-phase flows. PLIC reconstructs interfaces from a scalar field that…
Learning well-separated features in high-dimensional spaces, such as text or image embeddings, is crucial for many machine learning applications. Achieving such separation can be effectively accomplished through the dispersion of…
This paper is partly based on a lecture delivered by the author at the ERC workshop "Geometric Partial Differential Equations" held in Pisa in September 2012. What is presented here is an expanded version of that lecture.