# An Interpretive Constrained Linear Model for ResNet and MgNet

arXiv: https://arxiv.org/abs/2112.07441 · HTML: https://mlsanity.com/paper/2112.07441/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 56 references that are in the map.
- First posted on arXiv 14 Dec 2021.

## Builds on (most cited first, 30 of 35)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Aggregated Residual Transformations for Deep Neural Networks](https://mlsanity.com/paper/1611.05431/) (2016)
- [Deep Compression: Compressing Deep Neural Networks with Pruning, Trained Quantization and Huffman Coding](https://mlsanity.com/paper/1510.00149/) (2015)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Inception-v4, Inception-ResNet and the Impact of Residual Connections on Learning](https://mlsanity.com/paper/1602.07261/) (2016)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [FractalNet: Ultra-Deep Neural Networks without Residuals](https://mlsanity.com/paper/1605.07648/) (2016)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [What Can ResNet Learn Efficiently, Going Beyond Kernels?](https://mlsanity.com/paper/1905.10337/) (2019)
- [ResNet with one-neuron hidden layers is a Universal Approximator](https://mlsanity.com/paper/1806.10909/) (2018)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Why Do Deep Residual Networks Generalize Better than Deep Feedforward Networks? -- A Neural Tangent Kernel Perspective](https://mlsanity.com/paper/2002.06262/) (2020)
- [Multi-level Residual Networks from Dynamical Systems View](https://mlsanity.com/paper/1710.10348/) (2017)
- [MgNet: A Unified Framework of Multigrid and Convolutional Neural Network](https://mlsanity.com/paper/1901.10415/) (2019)
- [PolyNet: A Pursuit of Structural Diversity in Very Deep Networks](https://mlsanity.com/paper/1611.05725/) (2016)
- [Learning Deep ResNet Blocks Sequentially using Boosting Theory](https://mlsanity.com/paper/1706.04964/) (2017)
- [SCAN: A Scalable Neural Networks Framework Towards Compact and Efficient Models](https://mlsanity.com/paper/1906.03951/) (2019)
- [Kernel-Based Smoothness Analysis of Residual Networks](https://mlsanity.com/paper/2009.10008/) (2020)
- [Approximation Properties of Deep ReLU CNNs](https://mlsanity.com/paper/2109.00190/) (2021)

## Built on by (most cited first, 6 of 6)

- [MgNO: Efficient Parameterization of Linear Operators via Multigrid](https://mlsanity.com/paper/2310.19809/) (2023)
- [M2NO: An Efficient Multi-Resolution Operator Framework for Dynamic Multi-Scale PDE Solvers](https://mlsanity.com/paper/2406.04822/) (2024)
- [An Enhanced V-cycle MgNet Model for Operator Learning in Numerical Partial Differential Equations](https://mlsanity.com/paper/2302.00938/) (2023)
- [Self-composing neural operators for high-frequency and multiscale PDE surrogates](https://mlsanity.com/paper/2508.20650/) (2025)
- [MGiaD: Multigrid in all dimensions. Efficiency and robustness by coarsening in resolution and channel dimensions](https://mlsanity.com/paper/2211.05525/) (2022)
- [FV-MgNet: Fully Connected V-cycle MgNet for Interpretable Time Series Forecasting](https://mlsanity.com/paper/2302.00962/) (2023)

## Abstract (arXiv)

We propose a constrained linear data-feature-mapping model as an interpretable mathematical model for image classification using a convolutional neural network (CNN). From this viewpoint, we establish detailed connections between the traditional iterative schemes for linear systems and the architectures of the basic blocks of ResNet- and MgNet-type models. Using these connections, we present some modified ResNet models that compared with the original models have fewer parameters and yet can produce more accurate results, thereby demonstrating the validity of this constrained learning data-feature-mapping assumption. Based on this assumption, we further propose a general data-feature iterative scheme to show the rationality of MgNet. We also provide a systematic numerical study on MgNet to show its success and advantages in image classification problems and demonstrate its advantages in comparison with established networks.