arXiv research

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

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…

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

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…

The image of the branch set of a PL branched cover between PL $n$-manifolds is a simplicial $(n-2)$-complex. We demonstrate that the reverse implication also holds: an open and discrete map $f \colon \mathbb{S}^n \to \mathbb{S}^n$ with the image of the branch set contained in a simplicial $(n-2)$-complex is equivalent …

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.…

We demonstrate new applications of the trace embedding lemma to the study of piecewise-linear surfaces and the detection of exotic phenomena in dimension four. We provide infinitely many pairs of homeomorphic 4-manifolds $W$ and $W'$ homotopy equivalent to $S^2$ which have smooth structures distinguished by several for…

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…

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…

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 …

Let SL(n,Z) be the special linear group over integers and $M =S^r_1 \times S^r_2,T^r_1 \times S^r_2$ , or $T^r_0 \times S^r_1 \times S^r_2$, products of spheres and tori. We prove that any group action of SL(n,Z) on $M^r$ by diffeomorphims or piecewise linear homeomorphisms is trivial if $r<n-1$. This confirms a conjec…

Construct pseudo-Anosovs from expanding interval maps, reconciling Thurston's construction.

The study examines how gamma positivity and PL homeomorphism types affect simplicial spheres.

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.

Signature uniquely identifies piecewise linear surfaces up to thin homotopy.

PARC uses piecewise linear predictors for regression and classification.

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

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^{…

This article is one of three highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's but never published. Organizers of the Workshops in Geometric Topology (http://www.uwm.edu/~craigg/workshopgtt.htm) with the support of the National Science Foundation have facilitated the pr…

Complete description of BNSR invariants for Lodha-Moore groups, proving finiteness properties.

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

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…

New cones in 4D space found with minimal mass.

Classifies critical complexes for embedding in 3-sphere.

New proof shows efficient ReLU networks for piecewise linear functions.

Equivariant neural networks use symmetry to interpret complex data.

Paper proposes variational inference for piecewise-linear systems.

The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.

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

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 method uses DC functions for piecewise linear regression.

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.

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.

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-…

Simplifies PLNNs to interpretable models for better explainability.

A new definition for vector fields extends the Jacobi set concept.

Efficiently recovers piecewise linear functions from noisy samples.

We study piecewise linear co-dimension two embeddings of closed oriented manifolds in Euclidean space, and show that any such embedding can always be isotoped to be a closed braid as long as the ambient dimension is at most five, extending results of Alexander (in ambient dimension three), and Viro and independently Ka…

Unified piecewise-linear Ricci flows improve community detection.

Piecewise-linear virtual knots are discussed and classified up to edge index six.

We consider the generic regularized optimization problem $\hat{\mathsfβ}(λ)=\arg \min_βL({\sf{y}},X{\sfβ})+λJ({\sfβ})$. Efron, Hastie, Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407--499] have shown that for the LASSO--that is, if $L$ is squared error loss and $J(β)=\|β\|_1$ is the $\ell_1$ norm of $β$--the opti…

The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2].…

Solves asset allocation for investors with utility functions and limits.

We present a detailed analysis of the class of regression decision tree algorithms which employ a regulized piecewise-linear node-splitting criterion and have regularized linear models at the leaves. From a theoretic standpoint, based on Rademacher complexity framework, we present new high-probability upper bounds for …

SGD trains ReLU networks to implement piecewise linear maps with at most 3 knot points.