1,256 research outputs found
Wishart Mechanism for Differentially Private Principal Components Analysis
We propose a new input perturbation mechanism for publishing a covariance
matrix to achieve -differential privacy. Our mechanism uses a
Wishart distribution to generate matrix noise. In particular, We apply this
mechanism to principal component analysis. Our mechanism is able to keep the
positive semi-definiteness of the published covariance matrix. Thus, our
approach gives rise to a general publishing framework for input perturbation of
a symmetric positive semidefinite matrix. Moreover, compared with the classic
Laplace mechanism, our method has better utility guarantee. To the best of our
knowledge, Wishart mechanism is the best input perturbation approach for
-differentially private PCA. We also compare our work with
previous exponential mechanism algorithms in the literature and provide near
optimal bound while having more flexibility and less computational
intractability.Comment: A full version with technical proofs. Accepted to AAAI-1
The Noisy Power Method: A Meta Algorithm with Applications
We provide a new robust convergence analysis of the well-known power method
for computing the dominant singular vectors of a matrix that we call the noisy
power method. Our result characterizes the convergence behavior of the
algorithm when a significant amount noise is introduced after each
matrix-vector multiplication. The noisy power method can be seen as a
meta-algorithm that has recently found a number of important applications in a
broad range of machine learning problems including alternating minimization for
matrix completion, streaming principal component analysis (PCA), and
privacy-preserving spectral analysis. Our general analysis subsumes several
existing ad-hoc convergence bounds and resolves a number of open problems in
multiple applications including streaming PCA and privacy-preserving singular
vector computation.Comment: NIPS 201
Online and Differentially-Private Tensor Decomposition
In this paper, we resolve many of the key algorithmic questions regarding
robustness, memory efficiency, and differential privacy of tensor
decomposition. We propose simple variants of the tensor power method which
enjoy these strong properties. We present the first guarantees for online
tensor power method which has a linear memory requirement. Moreover, we present
a noise calibrated tensor power method with efficient privacy guarantees. At
the heart of all these guarantees lies a careful perturbation analysis derived
in this paper which improves up on the existing results significantly.Comment: 19 pages, 9 figures. To appear at the 30th Annual Conference on
Advances in Neural Information Processing Systems (NIPS 2016), to be held at
Barcelona, Spain. Fix small typos in proofs of Lemmas C.5 and C.
MVG Mechanism: Differential Privacy under Matrix-Valued Query
Differential privacy mechanism design has traditionally been tailored for a
scalar-valued query function. Although many mechanisms such as the Laplace and
Gaussian mechanisms can be extended to a matrix-valued query function by adding
i.i.d. noise to each element of the matrix, this method is often suboptimal as
it forfeits an opportunity to exploit the structural characteristics typically
associated with matrix analysis. To address this challenge, we propose a novel
differential privacy mechanism called the Matrix-Variate Gaussian (MVG)
mechanism, which adds a matrix-valued noise drawn from a matrix-variate
Gaussian distribution, and we rigorously prove that the MVG mechanism preserves
-differential privacy. Furthermore, we introduce the concept
of directional noise made possible by the design of the MVG mechanism.
Directional noise allows the impact of the noise on the utility of the
matrix-valued query function to be moderated. Finally, we experimentally
demonstrate the performance of our mechanism using three matrix-valued queries
on three privacy-sensitive datasets. We find that the MVG mechanism notably
outperforms four previous state-of-the-art approaches, and provides comparable
utility to the non-private baseline.Comment: Appeared in CCS'1
Beating Randomized Response on Incoherent Matrices
Computing accurate low rank approximations of large matrices is a fundamental
data mining task. In many applications however the matrix contains sensitive
information about individuals. In such case we would like to release a low rank
approximation that satisfies a strong privacy guarantee such as differential
privacy. Unfortunately, to date the best known algorithm for this task that
satisfies differential privacy is based on naive input perturbation or
randomized response: Each entry of the matrix is perturbed independently by a
sufficiently large random noise variable, a low rank approximation is then
computed on the resulting matrix.
We give (the first) significant improvements in accuracy over randomized
response under the natural and necessary assumption that the matrix has low
coherence. Our algorithm is also very efficient and finds a constant rank
approximation of an m x n matrix in time O(mn). Note that even generating the
noise matrix required for randomized response already requires time O(mn)
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