# Derivation of the Backpropagation Algorithm Based on Derivative Amplification Coefficients

arXiv: https://arxiv.org/abs/2102.04320 · HTML: https://mlsanity.com/paper/2102.04320/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 3 references that are in the map.
- First posted on arXiv 8 Feb 2021.

## Built on by (most cited first, 2 of 2)

- [Layer-Wise Partitioning and Merging for Efficient and Scalable Deep Learning](https://mlsanity.com/paper/2207.11019/) (2022)
- [Derivation of Back-propagation for Graph Convolutional Networks using Matrix Calculus and its Application to Explainable Artificial Intelligence](https://mlsanity.com/paper/2408.01408/) (2024)

## Abstract (arXiv)

The backpropagation algorithm for neural networks is widely felt hard to understand, despite the existence of some well-written explanations and/or derivations. This paper provides a new derivation of this algorithm based on the concept of derivative amplification coefficients. First proposed by this author for fully connected cascade networks, this concept is found to well carry over to conventional feedforward neural networks and it paves the way for the use of mathematical induction in establishing a key result that enables backpropagation for derivative amplification coefficients. Then we establish the connection between derivative amplification coefficients and error coefficients (commonly referred to as errors in the literature), and show that the same backpropagation procedure can be used for error coefficients. The entire derivation is thus rigorous, simple, and elegant.