# The Neural Echo: A Signal Processing Perspective for Understanding Neural Networks

arXiv: https://arxiv.org/abs/2608.04864 · HTML: https://mlsanity.com/paper/2608.04864/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 30 references that are in the map.
- First posted on arXiv 5 Aug 2026.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [The Limitations of Deep Learning in Adversarial Settings](https://mlsanity.com/paper/1511.07528/) (2015)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Full-Gradient Representation for Neural Network Visualization](https://mlsanity.com/paper/1905.00780/) (2019)
- [Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --](https://mlsanity.com/paper/2301.03362/) (2023)
- [The Filter Echo: A General Tool for Filter Visualisation](https://mlsanity.com/paper/2509.11932/) (2025)

## Abstract (arXiv)

We introduce the neural echo as a tool for understanding the behavior of neural networks. It generalizes the model-based concepts of impulse responses, diffusion echoes, and filter echoes to learning-based methods. It provides local, space-adaptive impulse responses and filter kernels for a neural network, its so-called echoes. These echoes depend on the input image and can be visualized to understand the learned dynamics of the network via an affine mapping. Neural echoes build a bridge from classical signal processing to modern explainable AI. They are very general and can be applied to both image-to-image and classification networks, with convolutional or fully connected structure, of feedforward or recurrent type, including modern transformer networks. Network differentiability is not required. In the differentiable case, neural echoes comprise concepts based on the network Jacobian, such as saliency maps and the analysis of adversarial perturbations, as special instances. As a simple blueprint to explain our framework, we derive neural echoes for the denoising convolutional neural network (DnCNN). Our experiments suggest that this network weights pixels based on their spatial and gray value distances. This not only clarifies its behavior, but also shows that it can reproduce key concepts of classical model-based denoisers such as bilateral filtering.