Expressive Power and Loss Surfaces of Deep Learning Models

TL;DR

This paper provides an intuitive geometric overview of deep learning's success, explores the expressive power of neural networks, and offers new insights into their loss surfaces, especially regarding multiplication neurons and complex mathematical perspectives.

Contribution

It introduces novel insights into the geometry and expressive capacity of deep models, emphasizing the role of multiplication neurons and interconnected loss surface analyses.

Findings

Deep neural networks carve out manifolds with multiplication neurons.

Connections between polynomial, matrix, spin glass, and complexity perspectives are established.

New insights into the geometry and loss surfaces of deep models are presented.

Abstract

The goals of this paper are two-fold. The first goal is to serve as an expository tutorial on the working of deep learning models which emphasizes geometrical intuition about the reasons for success of deep learning. The second goal is to complement the current results on the expressive power of deep learning models and their loss surfaces with novel insights and results. In particular, we describe how deep neural networks carve out manifolds especially when the multiplication neurons are introduced. Multiplication is used in dot products and the attention mechanism and it is employed in capsule networks and self-attention based transformers. We also describe how random polynomial, random matrix, spin glass and computational complexity perspectives on the loss surfaces are interconnected.