The Compact Support Neural Network

TL;DR

A novel neuron generalization that has the standard dot-product-based neuron and the radial basis function (RBF) neuron as two extreme cases of a shape parameter and proof that a neural network with such neurons has the universal approximation property means that it can approximate any continuous and integrable function with an arbitrary degree of accuracy.

Abstract

Neural networks are popular and useful in many fields, but they have the problem of giving high confidence responses for examples that are away from the training data. This makes the neural networks very confident in their prediction while making gross mistakes, thus limiting their reliability for safety-critical applications such as autonomous driving, space exploration, etc. This paper introduces a novel neuron generalization that has the standard dot-product-based neuron and the {\color{black} radial basis function (RBF)} neuron as two extreme cases of a shape parameter. Using a rectified linear unit (ReLU) as the activation function results in a novel neuron that has compact support, which means its output is zero outside a bounded domain. To address the difficulties in training the proposed neural network, it introduces a novel training method that takes a pretrained standard neural network that is fine-tuned while gradually increasing the shape parameter to the desired value. The theoretical findings of the paper are a bound on the gradient of the proposed neuron and a proof that a neural network with such neurons has the universal approximation property. This means that the network can approximate any continuous and integrable function with an arbitrary degree of accuracy. The experimental findings on standard benchmark datasets show that the proposed approach has smaller test errors than state-of-the-art competing methods and outperforms the competing methods in detecting out-of-distribution samples on two out of three datasets.

Figures (6)

• Figure S1: Left: 1D example. Comparison between RBF neuron ${\color{black} y=}\exp(-|x-2|^2)$ and compact support neurons ${\color{black} y=}f_\alpha(x,2,0,1)$from \ref{['eq:csn']} for $\alpha\in \{0,0.8,1\}$. Right: 2D example. The construction \ref{['eq:csn']} smoothly interpolates between a standard neuron ($\alpha=0$) and an RBF-type of neuron ($\alpha=1$). Shown are the decision boundaries for $f_\alpha(\mathbf{x},\mathbf{w},0,1)$ with $\mathbf{x}=(x_1,x_2)$, $\mathbf{w}=(0,2)$ for $\alpha\in\{0,0.1,0.5,0.8,1\}$ and the corresponding centers $\mathbf{w}/\alpha$ as "*".
• Figure S2: Left: a simple compact support neural network (CSNN), with the CSN layer described in \ref{['eq:csn_layer']}. Right: a CSNN-F with LeNet backbone, where all layers are trainable.
• Figure S3: The confidence map ($0.5$ for white and $1$ for black) of the trained CSNN on the moons dataset for different values of $\alpha\in[0,1]$.
• Figure S4: CSNN train and test errors, and AUROC for OOD detection vs. $\alpha$ for the moons data.
• Figure S5: Example of activation pattern domains for a regular NN and a CSNN ($\alpha=0.825$), and the resulting confidence map ($0.5$ for white and $1$ for black) for $\alpha=0.825$ for a 32 neuron 2-layer CSNN.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Theorem 3