Related papers: Multidimensional exact classes, smooth approximati…

We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain $X\subset \mathbb{R}^{2m-1}$, we find geometric significance in the function $\mathcal{M}_X(R) = \mathrm{mag}(R\cdot X)$. The function…

In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat $n$-manifold $M$, we show that the set of similarity classes of flat metrics on $M$ which occur as a cusp cross-section of a…

We prove matching direct and inverse theorems for uniform polynomial approximation with $A^*$ weights (a subclass of doubling weights suitable for approximation in the $L_\infty$ norm) having finitely many zeros and not too "rapidly…

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then…

In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes…

We study a model of machine teaching where the teacher mapping is constructed from a size function on both concepts and examples. The main question in machine teaching is the minimum number of examples needed for any concept, the so-called…

This current article aims to study a new subclass of meromorphic functions with positive coefficients by reconstructing a new operator in the punctured open disc. Also, some geometric properties are considered and investigated, such results…

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr,…

For any subgroup $G$ of the symmetric group $\mathcal{S}_n$ on $n$ symbols, we present results for the uniform $\mathcal{C}^k$ approximation of $G$-invariant functions by $G$-invariant polynomials. For the case of totally symmetric…

It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…

This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by $L^2$.…

Representation learning is currently a very hot topic in modern machine learning, mostly due to the great success of the deep learning methods. In particular low-dimensional representation which discriminates classes can not only enhance…

Manifold models provide low-dimensional representations that are useful for processing and analyzing data in a transformation-invariant way. In this paper, we study the problem of learning smooth pattern transformation manifolds from image…

We address the problem of inferring self-supervised dense semantic correspondences between objects in multi-object scenes. The method introduces learning of class-aware dense object descriptors by providing either unsupervised discrete…

This article introduces a novel, geometric approach for multi-manifold clustering (MMC), i.e. for clustering a collection of potentially intersecting, d-dimensional manifolds into the individual manifold components. We first compute a…

Neural networks are widely used as a model for classification in a large variety of tasks. Typically, a learnable transformation (i.e. the classifier) is placed at the end of such models returning a value for each class used for…

We consider the fundamental learning problem of estimating properties of distributions over large domains. Using a novel piecewise-polynomial approximation technique, we derive the first unified methodology for constructing sample- and…

We introduce a family of symmetric convex bodies called generalized ellipsoids of degree $d$ (GE-$d$s), with ellipsoids corresponding to the case of $d=0$. Generalized ellipsoids (GEs) retain many geometric, algebraic, and algorithmic…

The main motivation of this paper arises from the study of Carnot-Carath\'eodory spaces, where the class of 1-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including…

We study the class of 2-dimensional affine k-domains R satisfying ML(R) = k, where k is an arbitrary field of characteristic zero. In particular, we obtain the following result: Let R be a localization of a polynomial ring in finitely many…