Any Target Function Exists in a Neighborhood of Any Sufficiently Wide Random Network: A Geometrical Perspective
Machine Learning
2020-03-19 v2 Machine Learning
Abstract
It is known that any target function is realized in a sufficiently small neighborhood of any randomly connected deep network, provided the width (the number of neurons in a layer) is sufficiently large. There are sophisticated theories and discussions concerning this striking fact, but rigorous theories are very complicated. We give an elementary geometrical proof by using a simple model for the purpose of elucidating its structure. We show that high-dimensional geometry plays a magical role: When we project a high-dimensional sphere of radius 1 to a low-dimensional subspace, the uniform distribution over the sphere reduces to a Gaussian distribution of negligibly small covariances.
Keywords
Cite
@article{arxiv.2001.06931,
title = {Any Target Function Exists in a Neighborhood of Any Sufficiently Wide Random Network: A Geometrical Perspective},
author = {Shun-ichi Amari},
journal= {arXiv preprint arXiv:2001.06931},
year = {2020}
}