# Notes on: Olah, C. (2014): Neural networks, manifolds and topology

## Table of Contents

## Notation

## Definitions

Formally, an ambient isotopy between manifolds and is a continuous function such that each is a **homeomorphism** from to its range, is the identity function, and maps to . That is, continuously transitions from mapping to itself to mapping to .

## Overview

- Suggests that training NNs is iteratively mapping between manifolds (and if you assume the weight-matrix to be non-singular, this mapping is
**homeomorphic**) with the goal of obtaining a manifold in which the data is separable - Mentions that if we assume
**The Manifold Hypothesis**to be true, which says that natural data forms lower-dimensional manifolds in its embedding space, then the task of classification is fundamentally to separate a bunch of tangled manifolds.