# Convergence Analysis for Deep Sparse Coding via Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2408.05540 · HTML: https://mlsanity.com/paper/2408.05540/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 55 references that are in the map.
- First posted on arXiv 10 Aug 2024.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [YOLOv4: Optimal Speed and Accuracy of Object Detection](https://mlsanity.com/paper/2004.10934/) (2020)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [On the Relationship between Self-Attention and Convolutional Layers](https://mlsanity.com/paper/1911.03584/) (2019)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Activation Functions in Deep Learning: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2109.14545/) (2021)
- [Mish: A Self Regularized Non-Monotonic Activation Function](https://mlsanity.com/paper/1908.08681/) (2019)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds](https://mlsanity.com/paper/1808.10038/) (2018)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [Deep ReLU network approximation of functions on a manifold](https://mlsanity.com/paper/1908.00695/) (2019)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [Convolutional Neural Networks Analyzed via Convolutional Sparse Coding](https://mlsanity.com/paper/1607.08194/) (2016)
- [Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks](https://mlsanity.com/paper/2109.02832/) (2021)
- [Deep nets for local manifold learning](https://mlsanity.com/paper/1607.07110/) (2016)
- [Theory of Deep Convolutional Neural Networks III: Approximating Radial Functions](https://mlsanity.com/paper/2107.00896/) (2021)
- [On Multi-Layer Basis Pursuit, Efficient Algorithms and Convolutional Neural Networks](https://mlsanity.com/paper/1806.00701/) (2018)
- [Multi-Layer Convolutional Sparse Modeling: Pursuit and Dictionary Learning](https://mlsanity.com/paper/1708.08705/) (2017)
- [Depth Selection for Deep ReLU Nets in Feature Extraction and Generalization](https://mlsanity.com/paper/2004.00245/) (2020)

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

- [Sparse-Aware Neural Networks for Nonlinear Functionals: Mitigating the Exponential Dependence on Dimension](https://mlsanity.com/paper/2604.06774/) (2026)
- [Optimal Control with Natural Images: Efficient Reinforcement Learning using Overcomplete Sparse Codes](https://mlsanity.com/paper/2412.08893/) (2024)

## Abstract (arXiv)

In this work, we explore the intersection of sparse coding theory and deep learning to enhance our understanding of feature extraction capabilities in advanced neural network architectures. We begin by introducing a novel class of Deep Sparse Coding (DSC) models and establish a thorough theoretical analysis of their uniqueness and stability properties. By applying iterative algorithms to these DSC models, we derive convergence rates for convolutional neural networks (CNNs) in their ability to extract sparse features. This provides a strong theoretical foundation for the use of CNNs in sparse feature-learning tasks. We additionally extend this convergence analysis to more general neural network architectures, including those with diverse activation functions, as well as self-attention and transformer-based models. This broadens the applicability of our findings to a wide range of deep learning methods for the extraction of deep-sparse features. Inspired by the strong connection between sparse coding and CNNs, we also explore training strategies to encourage neural networks to learn sparser features. Through numerical experiments, we demonstrate the effectiveness of these approaches, providing valuable insight for the design of efficient and interpretable deep learning models.