arXiv research

Study Teichmüller dynamics and dilation tori properties, proving almost all vertical foliations are Morse-Smale.

Constructs commensurating actions for groups of piecewise transformations.

The center of a quotient group of piecewise linear homeomorphisms is trivial.

Neural networks can represent complex piecewise functions efficiently.

New algorithm reveals piecewise affine structure of neural networks.

BN refines local partition geometry in piecewise-affine networks during training.

Paper presents ABGD for efficient piecewise linear regression in high dimensions.

Open and discrete maps with specific branch set images are equivalent to PL branched covers.

Study introduces combinatorial criterion for quasi-isometry groups of Euclidean spaces.

Method identifies latent variables from high-dimensional data with piecewise affine mixing.

We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge posed by Thurston in dimension three and confirms a conjecture by Kwasik and L…

Paper develops algorithms for PWA systems with polynomial regret.

Study geometrically characterizes piecewise circular curves with decreasing curvature.

A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not necessarily proper. These systems can exhibit focus-focus singularities, which corresp…

Hilbert initiated the standpoint in foundations of mathematics. From this standpoint, we allow only a finite number of repetitions of elementary operations when we construct objects and morphisms. When we start from a subset of a Euclidean space. Then we assume that any element of the line has only a finite number of c…

Here are versions of the proofs of two classic theorems of combinatorial topology. The first is the result that piecewise linearly homeomorphic simplicial complexes are related by stellar moves. This is used in the proof, modelled on that of Pachner, of the second theorem. This states that moves from only a finite coll…

Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such complete affine structures when Sigma is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface Sigma, the deformation space identifies with…

Max-affine regression method converges linearly using GD and SGD.

Study growth patterns in random networks using i.i.d. perturbations.

We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information about the connectedness of interior and boundary sets, and on the dimensions and intersections of boundary sets. As an application, we descri…

Self-affine tiles homeomorphic to a ball proven for a specific digit set.

This paper uses linear rational splines for invertible modeling, offering a simpler inverse and similar costs.

Paper proposes a new method for SP with covariates using PADR and ERM.

PAR provides a flexible framework for quantization in optimization problems.

An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with holonomy consisting of affine transformati…

Analyzes limits of flat metrics on complex curves.

This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles needed to forma triangulated embedded orientable surface in R^d having P as its geo…

For Bezier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge in terms of Hausdorff distance. We prove that the exterior angles of control polygons under subdivision converge to 0 at the rate of $O(\sqrt{\frac{1}{2^i}})$, where $i$ is the number of subdivisions.…

New tiles in higher dimensions are shown to be homeomorphic to balls.

Exact LAD line fitting via PALB with linear scaling and speed.

The paper defines cocycles for positive Anosov representations and constructs affine actions with bounded fundamental domains.

The paper designs neural networks with assurance for controlling nonlinear systems.

PARC uses piecewise linear predictors for regression and classification.

Affine equivalence of half-translation surfaces via saddle connection graphs.

New method uses DC functions for piecewise linear regression.

New applications of trace embedding lemma show exotic 4-manifolds properties.

We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circ…

The paper provides results regarding the computational complexity of hybrid system identification. More precisely, we focus on the estimation of piecewise affine (PWA) maps from input-output data and analyze the complexity of computing a global minimizer of the error. Previous work showed that a global solution could b…

Paper extends understanding of deep network nonlinearities using vector quantization and statistical inference.

New EM algorithm improves deep generative network training.

A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic…

We prove Csorba's conjecture that the Lovász complex Hom(C_5,K_n) of graph multimorphisms from the 5-cycle C_5 to the complete graph K_n is Z/2Z-equivariantly homeomorphic to the Stiefel manifold, V(n-1,2), the space of (ordered) orthonormal 2-frames in R^{n-1}. The equivariant piecewise-linear topology that we need is…

The paper studies affine manifolds with linear foliations and their topological properties.

This paper studies certain embedded spheres in closed affine manifolds. For $n \geq 3$, we investigate the dome bodies in a closed affine $n$-manifold $M$ with its boundary homeomorphic to a sphere under the assumption that a developing map restricted to a component of $\partial\hat{M}$ is an embedding onto a strictly …

Efficiently finds sparse solutions to max-plus equations for convex regression.

3D self-affine tiles with specific digit sets have boundary homeomorphic to a 2-sphere.

Arguments on PL,(=piecewise linear) topology work over any ordered field in the same way as over the real field, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot). One of the most fundamental properties of definable sets is that a compact definable …

A Margulis spacetime is a complete flat affine Lorentzian 3-manifold with free fundamental group. Associated to $M$ is a noncompact complete hyperbolic surface $Σ$. We study double extensions of $π_1 (M) \cong π_1 (Σ)$ when $Σ$ is homeomorphic to a projective plane minus two discs. We classify proper actions of this do…