# Orthogonal and Idempotent Transformations for Learning Deep Neural Networks

arXiv: https://arxiv.org/abs/1707.05974 · HTML: https://mlsanity.com/paper/1707.05974/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2017.

## Built on by (most cited first, 1 of 1)

- [Revisiting Residual Connections: Orthogonal Updates for Stable and Efficient Deep Networks](https://mlsanity.com/paper/2505.11881/) (2025)

## Abstract (arXiv)

Identity transformations, used as skip-connections in residual networks, directly connect convolutional layers close to the input and those close to the output in deep neural networks, improving information flow and thus easing the training. In this paper, we introduce two alternative linear transforms, orthogonal transformation and idempotent transformation. According to the definition and property of orthogonal and idempotent matrices, the product of multiple orthogonal (same idempotent) matrices, used to form linear transformations, is equal to a single orthogonal (idempotent) matrix, resulting in that information flow is improved and the training is eased. One interesting point is that the success essentially stems from feature reuse and gradient reuse in forward and backward propagation for maintaining the information during flow and eliminating the gradient vanishing problem because of the express way through skip-connections. We empirically demonstrate the effectiveness of the proposed two transformations: similar performance in single-branch networks and even superior in multi-branch networks in comparison to identity transformations.