Bounds in piecewise linear topology

Dynamic Computational Topology for Piecewise Linear Curves

A piecewise linear (PL) approximation often serves as the graphics representation for a parametric curve. Algorithms for preserving correct topology for a single static image are available, but significant challenges remain to ensure correct topology when the PL curve is changing shape during synchronized visualization with an ongoing simulation, such as a molecule writhing over time. A tubular...

Piecewise Linear Topology, Evolutionary Algorithms, and Optimization Problems

Schemata theory, Markov chains, and statistical mechanics have been used to explain how evolutionary algorithms (EAs) work. Incremental success has been achieved with all of these methods, but each has been stymied by limitations related to its less-thanglobal view. We show that moving the question into topological space helps in the understanding of why EAs work. Purpose. In this paper we use ...

Nearly-tight VC-dimension bounds for piecewise linear neural networks

We prove new upper and lower bounds on the VC-dimension of deep neural networks with the ReLU activation function. These bounds are tight for almost the entire range of parameters. Letting W be the number of weights and L be the number of layers, we prove that the VC-dimension is O(WL log(W )), and provide examples with VC-dimension Ω(WL log(W/L)). This improves both the previously known upper ...

Almost Linear VC Dimension Bounds for Piecewise Polynomial Networks

We compute upper and lower bounds on the VC dimension and pseudo-dimension of feedforward neural networks composed of piecewise polynomial activation functions. We show that if the number of layers is fixed, then the VC dimension and pseudo-dimension grow as WlogW, where W is the number of parameters in the network. This result stands in opposition to the case where the number of layers is unbo...

Piecewise Constant Controlled Linear Fuzzy Systems