# Stochastic Backward Euler: An Implicit Gradient Descent Algorithm for k-means Clustering

arXiv: https://arxiv.org/abs/1710.07746 · HTML: https://mlsanity.com/paper/1710.07746/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2017.

## Built on by (most cited first, 3 of 3)

- [Proof-of-Learning: Definitions and Practice](https://mlsanity.com/paper/2103.05633/) (2021)
- [IM-BERT: Enhancing Robustness of BERT through the Implicit Euler Method](https://mlsanity.com/paper/2505.06889/) (2025)
- [Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks](https://mlsanity.com/paper/2303.01767/) (2023)

## Abstract (arXiv)

In this paper, we propose an implicit gradient descent algorithm for the classic k-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is carried over to the next gradient step. We draw connections between the proposed stochastic backward Euler and the recent entropy stochastic gradient descent (Entropy-SGD) for improving the training of deep neural networks. Numerical experiments on various synthetic and real datasets show that the proposed algorithm provides better clustering results compared to k-means algorithms in the sense that it decreased the objective function (the cluster) and is much more robust to initialization.