Nearly Optimal Private Convolution

We study computing the convolution of a private input $x$ with a public input $h$, while satisfying the guarantees of $(\epsilon, \delta)$-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 $(\epsilon, \delta)$-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