arXiv research

Equivariant neural networks use symmetry to interpret complex data.

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.

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 …

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

GraN-GAN normalizes gradients for better GAN performance.

We obtain a criterion for approximability by embeddings of piecewise linear maps of a circle to the plane, analogous to the one proved by Minc for maps of a segment to the plane. Theorem. Let S be a triangulation of a circle with s vertices. Let f be a simplicial map of the graph S to the plane. The map f is approximab…

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

The paper explores conditions for lifting maps between graphs to embeddings.

This work generalizes bounds on the number of linear regions in CPWL NNs.

We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map portions of each layer's input-space to the same output. In this way, deep models c…

New definition of regular points for PL functions on manifolds.

New method characterizes surface quadrilateral layouts as special immersions.

This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical vari…

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(Ω, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…

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

Label assignment problems with large state spaces are important tasks especially in computer vision. Often the pairwise interaction (or smoothness prior) between labels assigned at adjacent nodes (or pixels) can be described as a function of the label difference. Exact inference in such labeling tasks is still difficul…

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.

PARC uses piecewise linear predictors for regression and classification.

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

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

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

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.

A method for identifying NPWARX models with arbitrary domains using probabilistic mixture models.

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^{…

Introduces HTV to measure function complexity in learning schemes.

Paper proposes variational inference for piecewise-linear systems.

New GP model estimates piecewise continuous functions.

New method uses DC functions for piecewise linear regression.

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

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 …

Neural networks can represent complex piecewise functions efficiently.

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

The paper proves vanishing bounded cohomology for various groups.

Let $\widehat{\mathcal{C}}_{\mathbb{Z}}$ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that the natural map from the smooth knot concordance group $\mathcal{C}$ to $\wideha…

We study algebraic varieties of ReLU networks to understand their representable functions.

In exchange for large quantities of data and processing power, deep neural networks have yielded models that provide state of the art predication capabilities in many fields. However, a lack of strong guarantees on their behaviour have raised concerns over their use in safety-critical applications. A first step to unde…

Proposes adaptive ridge regression for functional linear models with piecewise shapes.

New proof shows efficient ReLU networks for piecewise linear functions.

The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.

Support vector machines (SVM) and other kernel techniques represent a family of powerful statistical classification methods with high accuracy and broad applicability. Because they use all or a significant portion of the training data, however, they can be slow, especially for large problems. Piecewise linear classifie…

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…

Deep networks realize complex mappings that are often understood by their locally linear behavior at or around points of interest. For example, we use the derivative of the mapping with respect to its inputs for sensitivity analysis, or to explain (obtain coordinate relevance for) a prediction. One key challenge is tha…

XGBoost is often presented as the algorithm that wins every ML competition. Surprisingly, this is true even though predictions are piecewise constant. This might be justified in high dimensional input spaces, but when the number of features is low, a piecewise linear model is likely to perform better. XGBoost was exten…

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