Related papers: Subdivision and spline spaces
Supervised dimensionality reduction has emerged as an important theme in the last decade. Despite the plethora of models and formulations, there is a lack of a simple model which aims to project the set of patterns into a space defined by…
We consider spline functions over simplicial meshes in $\RR^n$. We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of…
We introduce a numerical method for reconstructing a multidimensional surface using the gradient of the surface measured at some values of the coordinates. The method consists of defining a multidimensional spline function and minimizing…
We present a general framework for analyzing the complexity of subdivision-based algorithms whose tests are based on the sizes of regions and their distance to certain sets (often varieties) intrinsic to the problem under study. We call…
Vertical decomposition is a widely used general technique for decomposing the cells of arrangements of semi-algebraic sets in ${{\mathbb R}}^d$ into constant-complexity subcells. In this paper, we settle in the affirmative a few…
We build a rigorous bridge between deep networks (DNs) and approximation theory via spline functions and operators. Our key result is that a large class of DNs can be written as a composition of max-affine spline operators (MASOs), which…
We derive a divergence formula for a group of regularization methods with an L2 constraint. The formula is useful for regularization parameter selection, because it provides an unbiased estimate for the number of degrees of freedom. We…
We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments. This problem has important applications in several areas of numerical…
Given a graph whose edges are labeled by ideals of a commutative ring R with identity, a generalized spline is a vertex labeling by the elements of R such that the difference of the labels on adjacent vertices lies in the ideal associated…
A new family of combined subdivision schemes with one tension parameter is proposed by the interpolatory and approximating subdivision schemes. The displacement vectors between the points of interpolatory and approximating subdivision…
Detecting slender, overlapping structures remains a challenge in computational microscopy. While recent coordinate-based approaches improve detection, they often produce less accurate splines than pixel-based methods. We introduce a…
Converting quad meshes to smooth manifolds, guided subdivision offers a way to combine the good highlight line distributions of recent G-spline constructions with the refinability of subdivision surfaces. Specifically, we present a C2…
In this thesis, a new approach for constructing subdivision algorithms for generalized quadratic and cubic B-spline subdivision for subdivision surfaces and volumes is presented. First, a catalog of quality criteria for these subdivision…
This paper presents a general framework for calculating the dimension of spline spaces over arbitrary rectilinear partitions using the smoothing cofactor method. The approach extends existing dimension theory for polynomial splines over…
Several known constructions relate initial degenerations of projective toric varieties and Grassmannians to regular subdivisions of appropriate point configurations. We define a general framework which allows for partial generalizations of…
We provide a constructive proof of a face-to-face simplical partition of a d-dimensional space for arbitrary d by generalizing the idea of Sommerville, used to create space-filling tetrahedra out of triangular base, to any dimension. Each…
Spline functions have long been used in numerical solution of differential equations. Recently it revives as isogeometric analysis, which offers integration of finite element analysis and NURBS based CAD into a single unified process.…
This paper introduces a novel framework for constructing $C^r$ basis functions for polynomial spline spaces of degree $d$ over arbitrary planar polygonal partitions, overturning the belief that basis functions cannot be constructed on…
We give sublinear-time approximation algorithms for some optimization problems arising in machine learning, such as training linear classifiers and finding minimum enclosing balls. Our algorithms can be extended to some kernelized versions…
The method of constructing approximate solutions of the first boundary value problem for linear differential equations based on incomplete (even and odd) trigonometric splines is considered. The theoretical positions are illustrated by…