Deep Neural Networks as the Semi-classical Limit of Topological Quantum Neural Networks: The problem of generalisation

TL;DR

This work explores the paradigmatic case of the perceptron, which is implemented as the semiclassical limit of Topological Quantum Neural Networks, and applies a novel algorithm that obtains similar results to standard neural networks, but without the need for training.

Abstract

Deep Neural Networks miss a principled model of their operation. A novel framework for supervised learning based on Topological Quantum Field Theory that looks particularly well suited for implementation on quantum processors has been recently explored. We propose using this framework to understand the problem of generalisation in Deep Neural Networks. More specifically, in this approach, Deep Neural Networks are viewed as the semi-classical limit of Topological Quantum Neural Networks. A framework of this kind explains the overfitting behavior of Deep Neural Networks during the training step and the corresponding generalisation capabilities. We explore the paradigmatic case of the perceptron, which we implement as the semiclassical limit of Topological Quantum Neural Networks. We apply a novel algorithm we developed, showing that it obtains similar results to standard neural networks, but without the need for training (optimisation).

Figures (9)

• Figure 1: Traditional U-shaped risk curve, describing the trade-off between underfitting and overfitting, and double-descent risk curve, from belkin2019reconciling.
• Figure 2: A $2$-dimensional manifold (reversed pair of pants) with $1$-dimensional boundaries is mapped by a TQFT, $\mathcal{F}$, to its corresponding linear map. On top, the boundary consists of two circles, and therefore the domain of the corresponding linear map is a tensor product of vector spaces, while the bottom consists of a single circle, and the linear map has a single vector space as target.
• Figure 3: The empirical double-descent curve (red) is reproduced when both topological and metric complexity are taken into account.
• Figure 4: Possible paths in $xt$-plane.
• Figure 5: Predominant paths in the $h \rightarrow 0$ limit.