# Adapting Regularized Low Rank Models for Parallel Architectures

## TL;DR

This paper presents a novel smooth non-convex reformulation of regularized low-rank recovery models that leverages parallel architectures, significantly speeding up computations and enabling real-time applications like background subtraction and MRI analysis.

## Contribution

It introduces a Burer-Monteiro based reformulation that allows efficient first-order optimization of regularized low-rank models on parallel hardware, with convergence guarantees.

## Key findings

- Achieves an order-of-magnitude speedup over existing RPCA solvers on GPU.
- Enables real-time applications such as background subtraction and MRI analysis.
- Provides a convergence certificate and suboptimality bounds for the non-convex formulation.

## Abstract

We introduce a reformulation of regularized low-rank recovery models to take advantage of GPU, multiple CPU, and hybridized architectures. Low-rank recovery often involves nuclear-norm minimization through iterative thresholding of singular values. These models are slow to fit and difficult to parallelize because of their dependence on computing a singular value decomposition at each iteration. Regularized low-rank recovery models also incorporate non-smooth terms to separate structured components (e.g. sparse outliers) from the low-rank component, making these problems more difficult. Using Burer-Monteiro splitting and marginalization, we develop a smooth, non-convex formulation of regularized low-rank recovery models that can be fit with first-order solvers. We develop a computable certificate of convergence for this non-convex program, and use it to establish bounds on the suboptimality of any point. Using robust principal component analysis (RPCA) as an example, we include numerical experiments showing that this approach is an order-of-magnitude faster than existing RPCA solvers on the GPU. We also show that this acceleration allows new applications for RPCA, including real-time background subtraction and MR image analysis.