A Submodularity-based Agglomerative Clustering Algorithm for the Privacy Funnel
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For the privacy funnel (PF) problem, we propose an efficient iterative agglomerative clustering algorithm based on the minimization of the difference of submodular functions (IAC-MDSF). For a data curator that wants to share the data X correlated with the sensitive information S, the PF problem is to generate the sanitized data X̂ that maintains a specified utility/fidelity threshold on I(X; X̂) while minimizing the privacy leakage I(S; X̂). Our IAC-MDSF algorithm starts with the original alphabet X̂ := X and iteratively merges the elements in the current alphabet X̂ that minimizes the Lagrangian function I(S;X̂) - λ I(X;X̂) . We prove that the best merge in each iteration of IAC-MDSF can be searched efficiently over all subsets of X̂ by the existing MDSF algorithms. We show that the IAC-MDSF algorithm also applies to the information bottleneck (IB), a dual problem to PF. By varying the value of the Lagrangian multiplier λ, we obtain the experimental results on a heart disease data set in terms of the Pareto frontier: I(S;X̂) vs. - I(X;X̂). We show that our IAC-MDSF algorithm outperforms the existing iterative pairwise merge approaches for both PF and IB and is computationally much less complex.
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