# Gradient flows and randomised thresholding: sparse inversion and classification

arXiv: https://arxiv.org/abs/2203.11555 · HTML: https://mlsanity.com/paper/2203.11555/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 56 references that are in the map.
- First posted on arXiv 22 Mar 2022.

## Builds on (most cited first, 8 of 8)

- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [ProxSkip: Yes! Local Gradient Steps Provably Lead to Communication Acceleration! Finally!](https://mlsanity.com/paper/2202.09357/) (2022)
- [Analysis of Stochastic Gradient Descent in Continuous Time](https://mlsanity.com/paper/2004.07177/) (2020)
- [Binary Classification of Alzheimer Disease using sMRI Imaging modality and Deep Learning](https://mlsanity.com/paper/1809.06209/) (2018)
- [A Continuous-time Stochastic Gradient Descent Method for Continuous Data](https://mlsanity.com/paper/2112.03754/) (2021)
- [Solution Paths of Variational Regularization Methods for Inverse Problems](https://mlsanity.com/paper/1808.01783/) (2018)
- [An MBO scheme for clustering and semi-supervised clustering of signed networks](https://mlsanity.com/paper/1901.03091/) (2019)

## Built on by (most cited first, 1 of 1)

- [Losing momentum in continuous-time stochastic optimisation](https://mlsanity.com/paper/2209.03705/) (2022)

## Abstract (arXiv)

Sparse inversion and classification problems are ubiquitous in modern data science and imaging. They are often formulated as non-smooth minimisation problems. In sparse inversion, we minimise, e.g., the sum of a data fidelity term and an L1/LASSO regulariser. In classification, we consider, e.g., the sum of a data fidelity term and a non-smooth Ginzburg--Landau energy. Standard (sub)gradient descent methods have shown to be inefficient when approaching such problems. Splitting techniques are much more useful: here, the target function is partitioned into a sum of two subtarget functions -- each of which can be efficiently optimised. Splitting proceeds by performing optimisation steps alternately with respect to each of the two subtarget functions. In this work, we study splitting from a stochastic continuous-time perspective. Indeed, we define a differential inclusion that follows one of the two subtarget function's negative subdifferential at each point in time. The choice of the subtarget function is controlled by a binary continuous-time Markov process. The resulting dynamical system is a stochastic approximation of the underlying subgradient flow. We investigate this stochastic approximation for an L1-regularised sparse inversion flow and for a discrete Allen-Cahn equation minimising a Ginzburg--Landau energy. In both cases, we study the longtime behaviour of the stochastic dynamical system and its ability to approximate the underlying subgradient flow at any accuracy. We illustrate our theoretical findings in a simple sparse estimation problem and also in low- and high-dimensional classification problems.