# Quartic First-Order Methods for Low-Rank Minimization

## TL;DR

This paper introduces quartic first-order methods for low-rank matrix problems, leveraging non-Euclidean geometries to improve scalability and performance in matrix completion and factorization tasks.

## Contribution

It develops a novel class of algorithms based on quartic kernels and Gram kernels, enhancing efficiency and scalability for nonconvex low-rank minimization.

## Key findings

- Algorithms scale well to large problems
- Achieve state-of-the-art performance
- Improve numerical stability and efficiency

## Abstract

We study a generalized nonconvex Burer-Monteiro formulation for low-rank minimization problems. We use recent results on non-Euclidean first order methods to provide efficient and scalable algorithms. Our approach uses geometries induced by quartic kernels on matrix spaces; for unconstrained cases we introduce a novel family of Gram kernels that considerably improves numerical performances. Numerical experiments for Euclidean distance matrix completion and symmetric nonnegative matrix factorization show that our algorithms scale well and reach state of the art performance when compared to specialized methods.