# Projection Free Rank-Drop Steps

## TL;DR

This paper introduces a rank-drop method integrated with the Frank-Wolfe algorithm to maintain low-rank solutions in nuclear norm constrained problems, reducing computational costs.

## Contribution

It proposes a novel rank-drop step that decreases the rank during optimization, enhancing efficiency in projection-free algorithms like Frank-Wolfe.

## Key findings

- Rank-drop steps significantly reduce the solution rank.
- The method maintains feasibility and can be integrated into existing algorithms.
- Experimental results show improved efficiency over standard Frank-Wolfe.

## Abstract

The Frank-Wolfe (FW) algorithm has been widely used in solving nuclear norm constrained problems, since it does not require projections. However, FW often yields high rank intermediate iterates, which can be very expensive in time and space costs for large problems. To address this issue, we propose a rank-drop method for nuclear norm constrained problems. The goal is to generate descent steps that lead to rank decreases, maintaining low-rank solutions throughout the algorithm. Moreover, the optimization problems are constrained to ensure that the rank-drop step is also feasible and can be readily incorporated into a projection-free minimization method, e.g., Frank-Wolfe. We demonstrate that by incorporating rank-drop steps into the Frank-Wolfe algorithm, the rank of the solution is greatly reduced compared to the original Frank-Wolfe or its common variants.