The Flag Median and FlagIRLS

TL;DR

This paper introduces the flag median, a robust prototype for Grassmannian data, and the FlagIRLS algorithm for its efficient computation, improving clustering performance on various datasets including images and videos.

Contribution

The paper proposes the flag median and the FlagIRLS algorithm, offering a computationally efficient and robust method for Grassmannian prototypes, outperforming existing methods in clustering tasks.

Findings

FlagIRLS converges in 4 iterations on synthetic data.

Grassmannian LBG with flag median improves cluster purity by at least 10%.

Flag median is robust to outliers and enhances clustering quality.

Abstract

Finding prototypes (e.g., mean and median) for a dataset is central to a number of common machine learning algorithms. Subspaces have been shown to provide useful, robust representations for datasets of images, videos and more. Since subspaces correspond to points on a Grassmann manifold, one is led to consider the idea of a subspace prototype for a Grassmann-valued dataset. While a number of different subspace prototypes have been described, the calculation of some of these prototypes has proven to be computationally expensive while other prototypes are affected by outliers and produce highly imperfect clustering on noisy data. This work proposes a new subspace prototype, the flag median, and introduces the FlagIRLS algorithm for its calculation. We provide evidence that the flag median is robust to outliers and can be used effectively in algorithms like Linde-Buzo-Grey (LBG) to produce improved clusterings on Grassmannians. Numerical experiments include a synthetic dataset, the MNIST handwritten digits dataset, the Mind's Eye video dataset and the UCF YouTube action dataset. The flag median is compared the other leading algorithms for computing prototypes on the Grassmannian, namely, the $\ell_2$-median and to the flag mean. We find that using FlagIRLS to compute the flag median converges in $4$ iterations on a synthetic dataset. We also see that Grassmannian LBG with a codebook size of $20$ and using the flag median produces at least a $10\%$ improvement in cluster purity over Grassmannian LBG using the flag mean or $\ell_2$-median on the Mind's Eye dataset.