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For the privacy funnel (PF) problem, we propose an efficient iterativeagglomerative clustering algorithm based on the minimization of the differenceof submodular functions (IAC-MDSF). For a data curator that wants to share thedata $X$ correlated with the sensitive information $S$, the PF problem is togenerate the sanitized data $\hat{X}$ that maintains a specifiedutility/fidelity threshold on $I(X; \hat{X})$ while minimizing the privacyleakage $I(S; \hat{X})$. Our IAC-MDSF algorithm starts with the originalalphabet $\hat{\mathcal{X}} := \mathcal{X}$ and iteratively merges the elementsin the current alphabet $\hat{\mathcal{X}}$ that minimizes the Lagrangianfunction $ I(S;\hat{X}) - \lambda I(X;\hat{X}) $. We prove that the best mergein each iteration of IAC-MDSF can be searched efficiently over all subsets of$\hat{\mathcal{X}}$ by the existing MDSF algorithms. We show that the IAC-MDSFalgorithm also applies to the information bottleneck (IB), a dual problem toPF. By varying the value of the Lagrangian multiplier $\lambda$, we obtain theexperimental results on a heart disease data set in terms of the Paretofrontier: $ I(S;\hat{X})$ vs. $- I(X;\hat{X})$. We show that our IAC-MDSFalgorithm outperforms the existing iterative pairwise merge approaches for bothPF and IB and is computationally much less complex.

A Submodularity-based Agglomerative Clustering Algorithm for the Privacy Funnel