# Convergence of gradient flow for learning convolutional neural networks

arXiv: https://arxiv.org/abs/2601.08547 · HTML: https://mlsanity.com/paper/2601.08547/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 9 references that are in the map.
- First posted on arXiv 13 Jan 2026.

## Builds on (most cited first, 7 of 7)

- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizers](https://mlsanity.com/paper/1910.05505/) (2019)
- [Generalized Huber Loss for Robust Learning and its Efficient Minimization for a Robust Statistics](https://mlsanity.com/paper/2108.12627/) (2021)
- [Geometry of Linear Convolutional Networks](https://mlsanity.com/paper/2108.01538/) (2021)
- [Convergence of gradient descent for learning linear neural networks](https://mlsanity.com/paper/2108.02040/) (2021)
- [Function Space and Critical Points of Linear Convolutional Networks](https://mlsanity.com/paper/2304.05752/) (2023)
- [The Riemannian Geometry Associated to Gradient Flows of Linear Convolutional Networks](https://mlsanity.com/paper/2507.06367/) (2025)

## Abstract (arXiv)

Convolutional neural networks are widely used in imaging and image recognition. Learning such networks from training data leads to the minimization of a non-convex function. This makes the analysis of standard optimization methods such as variants of (stochastic) gradient descent challenging. In this article we study the simplified setting of linear convolutional networks. We show that the gradient flow (to be interpreted as an abstraction of gradient descent) applied to the empirical risk defined via certain loss functions including the square loss always converges to a critical point, under a mild condition on the training data.