Effective reconstruction of data perturbed by random projections

Random Projection (RP) has raised great concern among the research community of privacy-preserving data mining, due to its high efficiency and utility, e.g., keeping the euclidean distances among the data points. It was shown in [33] that, if the original data set composed of m attributes is multiplied by a mixing matrix of k × m(m > k) which is random and orthogonal on expectation, then the k series of perturbed data can be released for mining purposes. Given the data perturbed by RP and some necessary prior knowledge, to our knowledge, little work has been done in reconstructing the original data to recover some sensitive information. In this paper, we choose several typical scenarios in data mining with different assumptions on prior knowledge. For the cases that an attacker has full or zero knowledge of the mixing matrix R, respectively, we propose reconstruction methods based on Underdetermined Independent Component Analysis (UICA) if the attributes of the original data are mutually independent and sparse, and propose reconstruction methods based on Maximum A Posteriori (MAP) if the attributes of the original data are correlated and nonsparse. Simulation results show that our reconstructions achieve high recovery rates, and outperform the reconstructions based on Principal Component Analysis (PCA). Successful reconstructions essentially mean the leakage of privacy, so our work identify the possible risks of RP when it is used for data perturbations.