arXiv research

Neural networks can represent complex piecewise functions efficiently.

New GP model estimates piecewise continuous functions.

Neural network models improve survival analysis with reduced computation time.

New method uses DC functions for piecewise linear regression.

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

Deep neural networks with piecewise-polynomial activations can approximate smooth functions and their derivatives.

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

Global approximation for piecewise linear paths via signatures.

GraN-GAN normalizes gradients for better GAN performance.

New algorithm predicts piecewise regular functions online.

New proof shows efficient ReLU networks for piecewise linear functions.

A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This…

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

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

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

PARC uses piecewise linear predictors for regression and classification.

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

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…

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…

We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipsc…

We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqu…

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

This paper tackles discontinuous neural networks for better approximation of piecewise continuous functions.

Proposed by Donoho (1997), Dyadic CART is a nonparametric regression method which computes a globally optimal dyadic decision tree and fits piecewise constant functions in two dimensions. In this article we define and study Dyadic CART and a closely related estimator, namely Optimal Regression Tree (ORT), in the contex…

BN refines local partition geometry in piecewise-affine networks during training.

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…

Deep Jump Gaussian Processes model high-dimensional piecewise functions.

Solves asset allocation for investors with utility functions and limits.

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 …

We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and in dimension three and higher we assume a foliation condition. We make no assumption regarding con…

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

New algorithm reduces regret in online learning for piecewise continuous functions.

Method identifies latent variables from high-dimensional data with piecewise affine mixing.

We show that on a two-dimensional compact nontrapping Riemannian manifold with strictly convex boundary, a piecewise constant function can be recovered from its integrals over geodesics. We adapt the injectivity proof which uses variations through geodesics to recover the function and we improve this result when the ma…

For many large undirected models that arise in real-world applications, exact maximumlikelihood training is intractable, because it requires computing marginal distributions of the model. Conditional training is even more difficult, because the partition function depends not only on the parameters, but also on the obse…

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…

Given a piecewise linear (PL) function $p$ defined on an open subset of $\R^n$, one may construct by elementary means a unique polyhedron with multiplicities $\D(p)$ in the cotangent bundle $\R^n\times \R^{n*}$ representing the graph of the differential of $p$. Restricting to dimension 2, we show that any smooth functi…

Transformers struggle to approximate smooth functions, relying on piecewise constant approximations.

Equivariant neural networks use symmetry to interpret complex data.

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].…

Efficiently recovers piecewise linear functions from noisy samples.

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…

A method identifies abrupt changes in functions with fixed confidence under noisy feedback.

Hi-fi priors enhance BNNs by learning flexible activations.

We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geome…

We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^2$. As a model class, we consider the set $\mathcal{E}^β(\mathbb R^d)$ of possibly discontinuous piecewise $C^β$ functions $f : [-1/2, 1/2]^…

Given $iid$ observations from an unknown absolute continuous distribution defined on some domain $Ω$, we propose a nonparametric method to learn a piecewise constant function to approximate the underlying probability density function. Our density estimate is a piecewise constant function defined on a binary partition o…

This paper describes another extension of the Local Variance Gamma model originally proposed by P. Carr in 2008, and then further elaborated on by Carr and Nadtochiy, 2017 (CN2017), and Carr and Itkin, 2018 (CI2018). As compared with the latest version of the model developed in CI2018 and called the ELVG (the Expanded …