arXiv research

Signature uniquely identifies piecewise linear surfaces up to thin homotopy.

The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.

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

A triangulated piecewise-linear minimal surface in Euclidean 3-space defined using a variational characterization is critical for area amongst all continuous piecewise-linear variations with compact support that preserve the simplicial structure. We explicitly construct examples of such surfaces that are embedded and a…

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

We consider smooth isotropic immersions from the 2-dimensional torus into $R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed Lagrangian torus of $R^4$. We prove that such isotropic immersions can be approximated by arbitrarily $C^0$-close piecewise linear isotropic maps. If $n \geq 3$ the piece…

Tropical geometry and weighted lattices improve curve and surface fitting.

Paper connects surface shape analysis and unbalanced optimal transport.

In this paper, we introduce a parameterized discrete curvature ($α$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uni…

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

New method characterizes surface quadrilateral layouts as special immersions.

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…

A fundamental object in a hyperbolic 3-manifold M is its convex core C(M), defined as the smallest closed non-empty convex subset of M. We investigate the way the geometry of the boundary S of C(M) varies as we vary the hyperbolic metric of M. Thurston observed that the intrinsic metric of S is hyperbolic, and that its…

Universal triangulation for flat tori with 2434 triangles.

We consider Zimmer's program of lattice actions on surfaces by PL homomorphisms. It is proved that when the surface is not the torus or Klein bottle the action of any finite-index subgroup of SL(n,Z), n>4, (more generally for any 2-big lattice), factors through a finite group action. The proof is based on an establishm…

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

Simplified proof for dimension reduction of polygonal curves.

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. …

Global approximation for piecewise linear paths via signatures.

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…

Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson form converges to a piecewise linear form first defined by Kontsevich. The proof …

PARC uses piecewise linear predictors for regression and classification.

In this paper, we introduce a bordism category $\mathcal{C}_d^{PL}$ whose objects are bundles of closed $(d-1)$-dimensional piecewise linear manifolds and whose morphisms are bundles of $d$-dimensional piecewise linear cobordisms. In the main theorem of this article, we show that the classifying space $B\mathcal{C}_d^{…

We study strip deformations of convex cocompact hyperbolic surfaces, defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface. We prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation, in…

The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elem…

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

Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at the boundary of S perpendicularly are coordinates on the Teichmueller space T(S). We compute the Weil-Petersson Poisson structure on T(S) in this system of coordinates and we prove that i…

New cones in 4D space found with minimal mass.

New proof shows efficient ReLU networks for piecewise linear functions.

Crooked planes are piecewise linear surfaces that were introduced by Drumm in the early 1990s to construct fundamental domains for properly discontinuous actions of free groups on Minkowski 3-space. In a previous paper, we introduced analogues of these surfaces, called AdS crooked planes, in the 3-dimensional anti-de S…

Equivariant neural networks use symmetry to interpret complex data.

Paper proposes variational inference for piecewise-linear systems.

A hybrid model combines piecewise linear and neural components for interpretable predictions.

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in http://arxiv.org/abs/math.AG/0209253. The result is established with the help of the …

Proves minimum number of normals to curves in 3D space.

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…

New method uses DC functions for piecewise linear regression.

T-curves are piecewise linear curves which have been used with success since the beginning of the 1990's to construct new real algebraic curves with prescribed topology mainly on the real projective plane. In fact T-curves can be used on any real projective toric surface. We generalize here the construction of the latt…

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

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

SRNF framework extends surface distance to Lipschitz surfaces.

We reparametrize ReLU NNs as splines to understand their learning dynamics.

New proof for global rigidity of vertex scaling on polyhedral surfaces.

The paper studies which branched covers can be lifted to braided embeddings.

This paper concerns a method of selecting a subset of features for a sequential logit model. Tanaka and Nakagawa (2014) proposed a mixed integer quadratic optimization formulation for solving the problem based on a quadratic approximation of the logistic loss function. However, since there is a significant gap between …

Polytopes in high dimensions have at least 2n+4 normals.

Simplifies PLNNs to interpretable models for better explainability.

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-…