Multivariate microaggregation by iterative optimization

Abstract

Microaggregation is a well-known perturbative approach to publish personal or financial records while preserving the privacy of data subjects. Microaggregation is also a mechanism to realize the k-anonymity model for Privacy Preserving Data Publishing (PPDP). Microaggregation consists of two successive phases: partitioning the underlying records into small clusters with at least k records and aggregating the clustered records by a special kind of cluster statistic as a replacement. Optimal multivariate microaggregation has been shown to be NP-hard. Several heuristic approaches have been proposed in the literature. This paper presents an iterative optimization method based on the optimal solution of the microaggregation problem (IMHM). The method builds the groups based on constrained clustering and linear programming relaxation and fine-tunes the results within an integrated iterative approach. Experimental results on both synthetic and real-world data sets show that IMHM introduces less information loss for a given privacy parameter, and can be adopted for different real world applications.

Appendix: Proofs related to the ImprovedMHM function (Fig. 4)

Assume a group of records G with n members {X ₁,X ₂,…,X _n } in a d dimensional space. Let $C_{1}^{n}[l]= \sum_{j\in \{1,2,\ldots ,n\}}X_{j}[l]/n$ represents the lth dimension (1≤l≤d) of the centroid of G, where X _j [l] denotes the lth dimension of X _j . Based on Eq. (1), it is easy to verify that the SSE of the group can be computed by SSE(G)=SSE(G[1])+SSE(G[2])+⋯+SSE(G[d]), where G[l]={X ₁[l],X ₂[l],…,X _n [l]}.

Lemma 1

If a new record X _n+1 is added to a group G ₁, the SSE of the new group G ₂=G ₁∪{X _n+1} denoted by SSE ₂ can be calculated based on the calculated SSE before the addition (SSE ₁) in O(1).

Proof

Let ΔSSE=SSE ₂−SSE ₁=∑_1≤l≤d SSE ₂[l]−SSE ₁[l]. Based on Eq. (1), we can expand the ΔSSE[l]:

(3)

The new centroid can be computed simply:

$$ C_{1}^{n+1}[l]={\bigl(n.C_{1}^{n}[l]+X_{n+1}[l] \bigr)}/{(n+1)}. $$

(4)

So Eq. (3) reduces to:

(5)

These two updates require O(1) computations. □

Equations (4) and (5) are used in the ImprovedMHM function (lines 25–26 of Fig. 4) to update the SSE and centroid of a new group.

Lemma 2

If a record X _n+1 replaces a member in the group G, say X ₁, the SSE of the new group can be calculated in O(1).

Proof

We have $\mathit{SSE}_{2}[l]=\sum_{2\leq j\leq n+1}(X_{j}[l]-C_{2}^{n+1}[l])^{2}$. Let Δ[l]=X _n+1[l]−X ₁[l], so $C_{2}^{n+1}[l]=C_{1}^{n}[l]+\Delta [l]/n$ and the SSE can be updated incrementally:

(6)

Equation (6) is used in line 19 of Fig. 4, and requires O(1) computations. □