# Successive Affine Learning for Deep Neural Networks

arXiv: https://arxiv.org/abs/2305.07996 · HTML: https://mlsanity.com/paper/2305.07996/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 30 references that are in the map.
- First posted on arXiv 13 May 2023.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Deep Learning for Medical Image Analysis](https://mlsanity.com/paper/1708.08987/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Nonlinear Approximation and (Deep) ReLU Networks](https://mlsanity.com/paper/1905.02199/) (2019)
- [DeepPET: A deep encoder-decoder network for directly solving the PET reconstruction inverse problem](https://mlsanity.com/paper/1804.07851/) (2018)
- [Convergence of Deep ReLU Networks](https://mlsanity.com/paper/2107.12530/) (2021)
- [Convergence of Deep Convolutional Neural Networks](https://mlsanity.com/paper/2109.13542/) (2021)
- [Multi-Grade Deep Learning](https://mlsanity.com/paper/2302.00150/) (2023)
- [Sparse Deep Neural Network for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2207.13266/) (2022)

## Built on by (most cited first, 4 of 4)

- [Uniform Convergence of Deep Neural Networks with Lipschitz Continuous Activation Functions and Variable Widths](https://mlsanity.com/paper/2306.01692/) (2023)
- [Multi-Grade Deep Learning for Partial Differential Equations with Applications to the Burgers Equation](https://mlsanity.com/paper/2309.07401/) (2023)
- [Computational Advantages of Multi-Grade Deep Learning: Convergence Analysis and Performance Insights](https://mlsanity.com/paper/2507.20351/) (2025)
- [Multigrade Neural Network Approximation](https://mlsanity.com/paper/2601.16884/) (2026)

## Abstract (arXiv)

This paper introduces a successive affine learning (SAL) model for constructing deep neural networks (DNNs). Traditionally, a DNN is built by solving a non-convex optimization problem. It is often challenging to solve such a problem numerically due to its non-convexity and having a large number of layers. To address this challenge, inspired by the human education system, the multi-grade deep learning (MGDL) model was recently initiated by the author of this paper. The MGDL model learns a DNN in several grades, in each of which one constructs a shallow DNN consisting of a relatively small number of layers. The MGDL model still requires solving several non-convex optimization problems. The proposed SAL model mutates from the MGDL model. Noting that each layer of a DNN consists of an affine map followed by an activation function, we propose to learn the affine map by solving a quadratic/convex optimization problem which involves the activation function only it after the weight matrix and the bias vector for the current layer have been trained. In the context of function approximation, for a given function the SAL model generates an expansion of the function with adaptive basis functions in the form of DNNs. We establish the Pythagorean identity and the Parseval identity for the system generated by the SAL model. Moreover, we provide a convergence theorem of the SAL process in the sense that either it terminates after a finite number of grades or the norms of its optimal error functions strictly decrease to a limit as the grade number increases to infinity. Furthermore, we present numerical examples of proof of concept which demonstrate that the proposed SAL model significantly outperforms the traditional deep learning model.