11,093 research outputs found
Nearly Optimal Private Convolution
We study computing the convolution of a private input with a public input
, while satisfying the guarantees of -differential
privacy. Convolution is a fundamental operation, intimately related to Fourier
Transforms. In our setting, the private input may represent a time series of
sensitive events or a histogram of a database of confidential personal
information. Convolution then captures important primitives including linear
filtering, which is an essential tool in time series analysis, and aggregation
queries on projections of the data.
We give a nearly optimal algorithm for computing convolutions while
satisfying -differential privacy. Surprisingly, we follow
the simple strategy of adding independent Laplacian noise to each Fourier
coefficient and bounding the privacy loss using the composition theorem of
Dwork, Rothblum, and Vadhan. We derive a closed form expression for the optimal
noise to add to each Fourier coefficient using convex programming duality. Our
algorithm is very efficient -- it is essentially no more computationally
expensive than a Fast Fourier Transform.
To prove near optimality, we use the recent discrepancy lowerbounds of
Muthukrishnan and Nikolov and derive a spectral lower bound using a
characterization of discrepancy in terms of determinants
Recovering Velocity Distributions via Penalized Likelihood
Line-of-sight velocity distributions are crucial for unravelling the dynamics
of hot stellar systems. We present a new formalism based on penalized
likelihood for deriving such distributions from kinematical data, and evaluate
the performance of two algorithms that extract N(V) from absorption-line
spectra and from sets of individual velocities. Both algorithms are superior to
existing ones in that the solutions are nearly unbiased even when the data are
so poor that a great deal of smoothing is required. In addition, the
discrete-velocity algorithm is able to remove a known distribution of
measurement errors from the estimate of N(V). The formalism is used to recover
the velocity distribution of stars in five fields near the center of the
globular cluster Omega Centauri.Comment: 18 LATEX pages, 10 Postscript figures, uses AASTEX, epsf.sty.
Submitted to The Astronomical Journal, May 199
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