Generalization of an Upper Bound on the Number of Nodes Needed to Achieve Linear Separability

TL;DR

This paper derives an upper bound on the number of nodes needed in a two-hidden-layer neural network to achieve linear separability, based on data structure and activation functions, with empirical validation.

Contribution

It provides a new theoretical upper bound on network size for linear separability considering data structure and activation functions, extending previous results.

Findings

Upper bound depends on data structure and activation function properties.

For leaky ReLU, similar bounds hold under certain slope conditions.

Empirical results support the theoretical bounds.

Abstract

An important issue in neural network research is how to choose the number of nodes and layers such as to solve a classification problem. We provide new intuitions based on earlier results by An et al. (2015) by deriving an upper bound on the number of nodes in networks with two hidden layers such that linear separability can be achieved. Concretely, we show that if the data can be described in terms of N finite sets and the used activation function f is non-constant, increasing and has a left asymptote, we can derive how many nodes are needed to linearly separate these sets. This will be an upper bound that depends on the structure of the data. This structure can be analyzed using an algorithm. For the leaky rectified linear activation function, we prove separately that under some conditions on the slope, the same number of layers and nodes as for the aforementioned activation functions is sufficient. We empirically validate our claims.