We examine Euclidean distance-preserving data perturbation as a tool for privacy-preserving data mining. Such perturbations allow many important data mining algorithms e.g. hierarchical and k-means clustering), with only minor modification, to be applied to the perturbed data and produce exactly the same results as if applied to the original data. However, the issue of how well the privacy of the original data is preserved needs careful study. We engage in this study by assuming the role of an attacker armed with a small set of known original data tuples (inputs). Little work has been done examining this kind of attack when the number of known original tuples is less than the number of data dimensions. We focus on this important case, develop and rigorously analyze an attack that utilizes any number of known original tuples. The approach allows the attacker to estimate the original data tuple associated with each perturbed tuple and calculate the probability that the estimation results in a privacy breach. On a real 16-dimensional dataset, we show that the attacker, with 4 known original tuples, can estimate an original unknown tuple with less than 7 error with probability exceeding 0.8.
Ting @cite_10 considered left-multiplication by a randomly generated orthogonal matrix. However, they assume the original data tuples are rows rather than columns as we do. As a result, Euclidean distance between original data tuples is not preserved, but, sample mean and covariance are. If the original data arose as independent samples from multi-variate Gaussian distribution, then the perturbed data allows inferences to be drawn about this underlying distribution just as well as the original data. For all but small or very high-dimensional datasets, their approach is more resistant to prior knowledge attacks than Euclidean distance preserving perturbations. Their perturbation matrix is @math ( @math the number of original data tuples), much bigger than Euclidean distance preserving perturbation matrices, @math ( @math the number of entries in each original data tuple).
@cite_10
2003571173 2953076353 2539873326 2295296939
This paper considers the problem of recovering an unknown sparse p×p matrix X from an m×m matrix Y=AXB T , where A and B are known m×p matrices with m≪p. The main result shows that there exist constructions of the sketching matrices A and B so that even if X has O(p) nonzeros, it can be recovered exactly and efficiently using a convex program as long as these nonzeros are not concentrated in any single row column of X. Furthermore, it suffices for the size of Y (the sketch dimension) to scale as m = O(√(# nonzeros in X) × log p). The results also show that the recovery is robust and stable in the sense that if X is equal to a sparse matrix plus a perturbation, then the convex program we propose produces an approximation with accuracy proportional to the size of the perturbation. Unlike traditional results on sparse recovery, where the sensing matrix produces independent measurements, our sensing operator is highly constrained (it assumes a tensor product structure). Therefore, proving recovery guarantees require nonstandard techniques. Indeed, our approach relies on a novel result concerning tensor products of bipartite graphs, which may be of independent interest. This problem is motivated by the following application, among others. Consider a p×n data matrix D, consisting of n observations of p variables. Assume that the correlation matrix X:=DD T is (approximately) sparse in the sense that each of the p variables is significantly correlated with only a few others. Our results show that these significant correlations can be detected even if we have access to only a sketch of the data S=AD with A ∈ R m×p . This paper provides novel results for the recovery of signals from undersampled measurements based on analysis @math -minimization, when the analysis operator is given by a frame. We both provide so-called uniform and nonuniform recovery guarantees for cosparse (analysis-sparse) signals using Gaussian random measurement matrices. The nonuniform result relies on a recovery condition via tangent cones and the uniform recovery guarantee is based on an analysis version of the null space property. Examining these conditions for Gaussian random matrices leads to precise bounds on the number of measurements required for successful recovery. In the special case of standard sparsity, our result improves a bound due to Rudelson and Vershynin concerning the exact reconstruction of sparse signals from Gaussian measurements with respect to the constant and extends it to stability under passing to approximately sparse signals and to robustness under noise on the measurements. We prove that low-rank matrices can be recovered efficiently from a small number of measurements that are sampled from orbits of a certain matrix group. As a special case, our theory makes statements about the phase retrieval problem. Here, the task is to recover a vector given only the amplitudes of its inner product with a small number of vectors from an orbit. Variants of the group in question have appeared under different names in many areas of mathematics. In coding theory and quantum information, it is the complex Clifford group; in time-frequency analysis the oscillator group; and in mathematical physics the metaplectic group. It affords one particularly small and highly structured orbit that includes and generalizes the discrete Fourier basis: While the Fourier vectors have coefficients of constant modulus and phases that depend linearly on their index, the vectors in said orbit have phases with a quadratic dependence. In quantum information, the orbit is used extensively and is known as the set of stabilizer states. We argue that due to their rich geometric structure and their near-optimal recovery properties, stabilizer states form an ideal model for structured measurements for phase retrieval. Our results hold for @math measurements, where the oversampling factor k varies between @math and @math depending on the orbit. The reconstruction is stable towards both additive noise and deviations from the assumption of low rank. If the matrices of interest are in addition positive semidefinite, reconstruction may be performed by a simple constrained least squares regression. Our proof methods could be adapted to cover orbits of other groups. In statistics and machine learning, people are often interested in the eigenvectors (or singular vectors) of certain matrices (e.g. covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. One usually employs Davis-Kahan @math theorem to bound the difference between the eigenvectors of a matrix @math and those of a perturbed matrix @math , in terms of @math norm. In this paper, we prove that when @math is a low-rank and incoherent matrix, the @math norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of @math or @math for left and right vectors, where @math and @math are the matrix dimensions. The power of this new perturbation result is shown in robust covariance estimation, particularly when random variables have heavy tails. There, we propose new robust covariance estimators and establish their asymptotic properties using the newly developed perturbation bound. Our theoretical results are verified through extensive numerical experiments.