arXiv research

This paper studies PL cobordism categories and their homotopy types.

First explicit isometric immersion of a flat Klein bottle in 3D space.

Smooth symplectic manifolds can be approximated by PL symplectic manifolds.

This is a survey paper where we expose the Kirby--Siebenmann results on classification of PL structures on topological manifolds and, in particular, the homotopy equivalence TOP/PL=K(Z/2.3) and the Hauptvermutung for manifolds.

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…

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

New definition of regular points for PL functions on manifolds.

Study geometrically characterizes piecewise circular curves with decreasing curvature.

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…

New knot concordance invariants from sl(n) cohomology.

LinXGBoost extends XGBoost for better regression of piecewise linear functions.

Study improves Morse-Smale regression for actuarial science using various machine learning algorithms.

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

Piecewise normalizing flows improve multi-modal distribution modeling.

New topological complexity measures for neural networks.

This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field. On a given set of piecewise linear spaces we define and discuss (normalized) Ricci flows. …

PARC uses piecewise linear predictors for regression and classification.

Global approximation for piecewise linear paths via signatures.

The paper approximates smooth isotropic surfaces with piecewise linear ones.

New algorithm reveals piecewise affine structure of neural networks.

Piecewise-linear regression trees improve tree-based regression with theoretical and practical benefits.

Investigates stability of piecewise flat Ricci flow using analysis and simulations.

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 …

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…

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…

Piecewise linear activations create many spurious local minima in neural networks.

Simplicial, piecewise-flat discretizations of manifolds provide a clear path towards curvature analysis on discrete geometries and for solutions of PDE's on manifolds of complex topologies. In this manuscript we review and expand on discrete exterior calculus methods using hybrid domains. We then analyze the geometric …

Signature uniquely identifies piecewise linear surfaces up to thin homotopy.

The paper extends a variance gamma model to quadratic functions, reducing arbitrage and computational costs.

New method characterizes surface quadrilateral layouts as special immersions.

The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or ``geometric'' knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In…

Computational topology shows knots can converge to stick knots.

New approach treats neural networks with piecewise linear activations using tropical geometry.

We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities $S^{n-2}\subset S^n$, $n\geq 5$, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement…

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.…

GraN-GAN normalizes gradients for better GAN performance.

Paper proposes variational inference for piecewise-linear systems.

Smooths local volatility model calibration with piecewise linear variance.

New GP model estimates piecewise continuous functions.

A new invariant captures geometric features of circle embeddings.

New method uses DC functions for piecewise linear regression.

A new complexity measure for neural networks improves upon classical methods.

Equivariant neural networks use symmetry to interpret complex data.

Embeddings in 5D can be isotoped to closed braids.

Piecewise Euclidean structures (identified solid Euclidean polyhedra) on topological 3-dimensional manifolds and pseudo-manifolds are constructed so that they admit pseudo-foliations, a generalized type of foliation. The construction of non-manifold point neighborhoods is done to preserve as much of the geometric, and …

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

Optimizes piecewise local-linear approximations for global model understanding.