GIST: Greedy Independent Set Thresholding for Max-Min Diversification with Submodular Utility

This work studies a novel subset selection problem called max-min diversification with monotone submodular utility (MDMS), which has a wide range of applications in machine learning, e.g., data sampling and feature selection. Given a set of points in a metric space, the goal of MDMS is to maximize $f(S)=g(S)+\lambda \cdot \text{div}(S)$ subject to a cardinality constraint $|S|\le k$, where $g(S)$ is a monotone submodular function and $\text{div}(S)={\min }_{u,v\in S:u\ne v}\text{dist}(u,v)$ is the max-min diversity objective.

We propose the GIST algorithm, which gives a $\frac{1}{2}$-approximation guarantee for MDMS by approximating a series of maximum independent set problems with a bicriteria greedy algorithm. We also prove that it is NP-hard to approximate within a factor of $0.5584$.

Finally, we show in our empirical study that GIST outperforms state-of-the-art benchmarks for a single-shot data sampling task on ImageNet.