GIST: Greedy Independent Set Thresholding for Max-Min Diversification with Submodular Utility
Matthew Fahrbach, Srikumar Ramalingam, Morteza Zadimoghaddam, Sara Ahmadian, Gui Citovsky, Giulia DeSalvo
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 functionand$\text{div}(S) = \min_{u,v \in S : u \ne v} \text{dist}(u,v)$ is the *max-min diversity* objective.We propose the `GIST` algorithm, which gives a $\frac{1}{2}$-approximation guarantee for MDMSby 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 benchmarksfor a single-shot data sampling task on ImageNet.