# Statistical Learning and Inverse Problems: A Stochastic Gradient Approach

arXiv: https://arxiv.org/abs/2209.14967 · HTML: https://mlsanity.com/paper/2209.14967/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 34 references that are in the map.
- First posted on arXiv 29 Sep 2022.

## Builds on (most cited first, 9 of 9)

- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [Functional Regression](https://mlsanity.com/paper/1406.4068/) (2014)
- [Deep Learning Methods for Solving Linear Inverse Problems: Research Directions and Paradigms](https://mlsanity.com/paper/2007.13290/) (2020)
- [On the Convergence of Stochastic Gradient Descent for Nonlinear Ill-Posed Problems](https://mlsanity.com/paper/1907.03132/) (2019)
- [On the Saturation Phenomenon of Stochastic Gradient Descent for Linear Inverse Problems](https://mlsanity.com/paper/2010.10916/) (2020)
- [Functional Classification of Bitcoin Addresses](https://mlsanity.com/paper/2202.12019/) (2022)

## Built on by (most cited first, 5 of 5)

- [Nonparametric Instrumental Variable Regression through Stochastic Approximate Gradients](https://mlsanity.com/paper/2402.05639/) (2024)
- [Controlling the Flow: Stability and Convergence for Stochastic Gradient Descent with Decaying Regularization](https://mlsanity.com/paper/2505.11434/) (2025)
- [Random Gradient-Free Optimization in Infinite Dimensional Spaces](https://mlsanity.com/paper/2512.20566/) (2025)
- [Functional Gradient Descent with Adaptive Representations](https://mlsanity.com/paper/2606.16926/) (2026)
- [Stochastic Gradient Langevin Dynamics Based on Quantization with Increasing Resolution](https://mlsanity.com/paper/2305.18864/) (2023)

## Abstract (arXiv)

Inverse problems are paramount in Science and Engineering. In this paper, we consider the setup of Statistical Inverse Problem (SIP) and demonstrate how Stochastic Gradient Descent (SGD) algorithms can be used in the linear SIP setting. We provide consistency and finite sample bounds for the excess risk. We also propose a modification for the SGD algorithm where we leverage machine learning methods to smooth the stochastic gradients and improve empirical performance. We exemplify the algorithm in a setting of great interest nowadays: the Functional Linear Regression model. In this case we consider a synthetic data example and examples with a real data classification problem.