arXiv research

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

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

Paper proposes algorithms to accurately identify breakpoints in piecewise regression.

New knots share same Upsilon invariant despite different Alexander polynomials.

New algorithm reduces dynamic regret for noisy gradient feedback with piecewise polynomial comparators.

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…

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…

Piecewise polynomial interpolation-based gradient descent reduces oracle complexity for smooth loss functions.

New algorithm predicts piecewise regular functions online.

Sample- and computationally-efficient distribution estimation is a fundamental tenet in statistics and machine learning. We present SURF, an algorithm for approximating distributions by piecewise polynomials. SURF is: simple, replacing prior complex optimization techniques by straight-forward {empirical probability} ap…

The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.

Efficiently finds sparse solutions to max-plus equations for convex regression.

We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is $τ$-close (in total variation distance) to an unknown probability distribution $…

Paper develops algorithms for PWA systems with polynomial regret.

The study describes a cell structure for multisets in a rectangle.

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…

Constructs finite element spaces for $(p,q)$-forms, excluding one subspace.

In many applications, data is collected in batches, some of which are corrupt or even adversarial. Recent work derived optimal robust algorithms for estimating discrete distributions in this setting. We consider a general framework of robust learning from batches, and determine the limits of both classification and dis…

Wide networks with polynomial activations have proven asymptotic behavior.

The paper studies geometric structures of polynomial spaces.

The paper analyzes a simple neural network model with algebraic methods.

ParamBoost uses gradient boosting to create interpretable non-linear models with constraints.

We consider an appoximation of a catenoid constructed from "odd" truncated cones that maintains minimality in a certain sense. Thorough this procedure, we obtain a discrete curve approximating a catenary by exploiting the fact that it is the function that generates a catenoid. In this investigation, the theory of the G…

Quantum Monte Carlo speeds up option pricing for complex payoff functions.

The paper approximates Einstein tensor using finite elements.

Existing works on "black-box" model interpretation use local-linear approximations to explain the predictions made for each data instance in terms of the importance assigned to the different features for arriving at the prediction. These works provide instancewise explanations and thus give a local view of the model. T…

We study additive models built with trend filtering, i.e., additive models whose components are each regularized by the (discrete) total variation of their $k$th (discrete) derivative, for a chosen integer $k \geq 0$. This results in $k$th degree piecewise polynomial components, (e.g., $k=0$ gives piecewise constant co…

This note is an addendum to our earlier work \cite{humi}. In \cite{humi}, we studied a Hamiltonian action for a generalized Calabi-Yau manifold and showed that the Duistermaat-Heckman theorem holds. The purpose of this note is to show that the density function of the Duistermaa-Heckman measure is a piecewise polynomial…

This technical note extends recent results on the computational complexity of globally minimizing the error of piecewise-affine models to the related problem of minimizing the error of switching linear regression models. In particular, we show that, on the one hand the problem is NP-hard, but on the other hand, it admi…

Hard problem of learning simple generative models from i.i.d. samples.

PolyLUT uses polynomials to reduce FPGA latency.

The paper provides results regarding the computational complexity of hybrid system identification. More precisely, we focus on the estimation of piecewise affine (PWA) maps from input-output data and analyze the complexity of computing a global minimizer of the error. Previous work showed that a global solution could b…

In this article we give an explicit algorithm which will determine, in a discrete and computable way, whether a finite piecewise Euclidean complex is non-positively curved. In particular, given such a complex we show how to define a boolean combination of polynomial equations and inequalities in real variables, i.e. a …

New method efficiently interpolates nonparametric density estimators.

In this paper, we investigate adaptive nonlinear regression and introduce tree based piecewise linear regression algorithms that are highly efficient and provide significantly improved performance with guaranteed upper bounds in an individual sequence manner. We use a tree notion in order to partition the space of regr…

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…

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

We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of…

A derivation of the Cesàro-Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-linear boundaries. Several results are obtained that relate the coefficients, $b_i$, in the Shephard-…

The Immersed Boundary (IB) method is a widely-used numerical methodology for the simulation of fluid-structure interaction problems. The IB method utilizes an Eulerian discretization for the fluid equations of motion while maintaining a Lagrangian representation of structural objects. Operators are defined for transmit…

Neural networks can represent complex piecewise functions efficiently.

Theorem proves integrability for piecewise-smooth distributions.

Normalizing flows attempt to model an arbitrary probability distribution through a set of invertible mappings. These transformations are required to achieve a tractable Jacobian determinant that can be used in high-dimensional scenarios. The first normalizing flow designs used coupling layer mappings built upon affine …

Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.

For applications in computing, Bezier curves are pervasive and are defined by a piecewise linear curve L which is embedded in R^3 and yields a smooth polynomial curve C embedded in R^3. It is of interest to understand when L and C have the same embeddings. One class of counterexamples is shown for L being unknotted, wh…

Let G be a connected compact Lie group acting on a manifold M and let D be a transversally elliptic operator on M. The multiplicity of the index of D is a function on the set of irreducible representations of G. Let T be a maximal torus of G with Lie algebra Lie(T). We construct a finite number of piecewise polynomial …

Study geometrically characterizes piecewise circular curves with decreasing curvature.

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