arXiv research

A new tree-based model for multivariate responses interprets piecewise linear regimes.

New method samples from piecewise smooth distributions using Hamiltonian Monte Carlo.

Study identifies change points in piecewise constant reward functions with fixed exploration budget.

PAR provides a flexible framework for quantization in optimization problems.

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

Study shows efficient neural network approach for stochastic bandits.

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

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.

Exact solutions reveal how unbalanced initializations promote rapid feature learning in neural networks.

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

New algorithm reveals piecewise affine structure of neural networks.

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…

Unified framework detects change-points and estimates parameters in nonlinear systems with regime switching.

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

This paper introduces a novel mixture model-based approach for simultaneous clustering and optimal segmentation of functional data which are curves presenting regime changes. The proposed model consists in a finite mixture of piecewise polynomial regression models. Each piecewise polynomial regression model is associat…

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.

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^{…

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

GraN-GAN normalizes gradients for better GAN performance.

Paper proposes variational inference for piecewise-linear systems.

New GP model estimates piecewise continuous functions.

New method uses DC functions for piecewise linear regression.

Equivariant neural networks use symmetry to interpret complex data.

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 …

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

Neural networks can represent complex piecewise functions efficiently.

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.

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

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.

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

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…

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…

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…

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…

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 …

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

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-…

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

We present a new, unifying approach following some recent developments on the complexity of neural networks with piecewise linear activations. We treat neural network layers with piecewise linear activations as tropical polynomials, which generalize polynomials in the so-called $(\max, +)$ or tropical algebra, with pos…