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Interactive exploration of large, multidimensional datasets plays a very important role in various scientific fields. It makes it possible not only to identify important structural features and forms, such as clusters of vertices and their…
The local a priori estimate for the finite element approximation is essential for underlying the local and parallel technique. It is well known that the constant coefficients in the inequality is independent of the mesh size. But it is not…
We study special subvarieties, i.e., subvarieties containing a dense subset of CM points, of the moduli space $A_5$ of principally polarized abelian varieties of dimension five, generically contained in the locus of intermediate Jacobians…
We study geometric and topological properties of the image of a smooth submanifold of $\mathbb{R}^{n}$ under a bi-Lipschitz map to $\mathbb{R}^{m}$. In particular, we characterize how the dimension, diameter, volume, and reach of the…
Self-Organizing Maps (SOMs, Kohonen networks) belong to neural network models of the unsupervised class. In this paper, we present the generalized setup for non-Euclidean SOMs. Most data analysts take it for granted to use some subregions…
Multicuts enable to conveniently represent discrete graphical models for unsupervised and supervised image segmentation, in the case of local energy functions that exhibit symmetries. The basic Potts model and natural extensions thereof to…
Representation learning for graphs enables the application of standard machine learning algorithms and data analysis tools to graph data. Replacing discrete unordered objects such as graph nodes by real-valued vectors is at the heart of…
Multivariate spatial modeling is key to understanding the behavior of materials downstream in a mining operation. The ore recovery depends on the mineralogical composition, which needs to be properly captured by the model to allow for good…
We analyse new exact enumeration data for self-avoiding polygons, counted by perimeter and area on the square, triangular and hexagonal lattices. In extending earlier analyses, we focus on the perimeter moments in the vicinity of the…
We consider manifolds whose transition maps are restrictions of polynomial mappings $\mathbb{R}^n\to\mathbb{R}^n$, and use them to give an equivalent statement of the Jacobian conjecture over the real field.
While existing mathematical descriptions can accurately account for phenomena at microscopic scales (e.g. molecular dynamics), these are often high-dimensional, stochastic and their applicability over macroscopic time scales of physical…
Bayes spaces were initially designed to provide a geometric framework for the modeling and analysis of distributional data. It has recently come to light that this methodology can be exploited to provide an orthogonal decomposition of…
We investigate the ability of a local bi-orthogonal decomposition to build texture segmentation of images. Using the structures associated to the local decomposition of the image independent row and columns we perform a segmentation, where…
A well-known open problem asks whether every bi-Lipschitz homeomorphism of $\mathbb{R}^d$ factors as a composition of mappings of small distortion. We show that every bi-Lipschitz embedding of the unit cube $[0,1]^d$ into $\mathbb{R}^d$…
In this note we investigate the connection between polylogarithms on curves and abelian schemes. The main result shows that the polylogarithm on the abelian scheme can be obtained as the push-forward of the polylogarithm on a suitable…
Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian scheme of Y, homological invariants derived from logarithmic differential forms along Y, and their relationship with the property that Y is a…
Low-dimensional embeddings (LDEs) of high-dimensional data are ubiquitous in science and engineering. They allow us to quickly understand the main properties of the data, identify outliers and processing errors, and inform the next steps of…
Manifold learning builds on the "manifold hypothesis," which posits that data in high-dimensional datasets are drawn from lower-dimensional manifolds. Current tools generate global embeddings of data, rather than the local maps used to…
Urban evolution processes occur at different scales, with intricate interactions between levels and relatively distinct type of processes. To what extent actual urban dynamics include an actual strong coupling between scales, in the sense…
We extend the Bayesian Information Criterion (BIC), an asymptotic approximation for the marginal likelihood, to Bayesian networks with hidden variables. This approximation can be used to select models given large samples of data. The…