Deep Layer-wise Networks Have Closed-Form Weights

TL;DR

This paper proves that layer-wise neural networks can have a closed-form solution for their weights using Kernel Mean Embedding, which guides convergence towards an optimal kernel for classification.

Contribution

It introduces a closed-form solution for layer-wise networks' weights using Kernel Mean Embedding and addresses when to stop adding layers.

Findings

Kernel Mean Embedding provides the closed-form weights.

Networks converge towards the Neural Indicator Kernel.

The method enhances understanding of layer-wise training dynamics.

Abstract

There is currently a debate within the neuroscience community over the likelihood of the brain performing backpropagation (BP). To better mimic the brain, training a network \textit{one layer at a time} with only a "single forward pass" has been proposed as an alternative to bypass BP; we refer to these networks as "layer-wise" networks. We continue the work on layer-wise networks by answering two outstanding questions. First, $\textit{do they have a closed-form solution?}$ Second, $\textit{how do we know when to stop adding more layers?}$ This work proves that the Kernel Mean Embedding is the closed-form weight that achieves the network global optimum while driving these networks to converge towards a highly desirable kernel for classification; we call it the $\textit{Neural Indicator Kernel}$.