The Optimal Mechanism in Differential Privacy

We derive the optimal $\epsilon$-differentially private mechanism for single real-valued query function under a very general utility-maximization (or cost-minimization) framework. The class of noise probability distributions in the optimal mechanism has {\em staircase-shaped} probability density functions which are symmetric (around the origin), monotonically decreasing and geometrically decaying. The staircase mechanism can be viewed as a {\em geometric mixture of uniform probability distributions}, providing a simple algorithmic description for the mechanism. Furthermore, the staircase mechanism naturally generalizes to discrete query output settings as well as more abstract settings. We explicitly derive the optimal noise probability distributions with minimum expectation of noise amplitude and power. Comparing the optimal performances with those of the Laplacian mechanism, we show that in the high privacy regime ($\epsilon$ is small), Laplacian mechanism is asymptotically optimal as $\epsilon \to 0$; in the low privacy regime ($\epsilon$ is large), the minimum expectation of noise amplitude and minimum noise power are $\Theta(\Delta e^{-\frac{\epsilon}{2}})$ and $\Theta(\Delta^2 e^{-\frac{2\epsilon}{3}})$ as $\epsilon \to +\infty$, while the expectation of noise amplitude and power using the Laplacian mechanism are $\frac{\Delta}{\epsilon}$ and $\frac{2\Delta^2}{\epsilon^2}$, where $\Delta$ is the sensitivity of the query function. We conclude that the gains are more pronounced in the low privacy regime.Comment: 40 pages, 5 figures. Part of this work was presented in DIMACS Workshop on Recent Work on Differential Privacy across Computer Science, October 24 - 26, 201

Phase Compensation Enhancement of Photon Pair Entanglement Generated from Biexciton Decays in Quantum Dots

Exciton fine-structure splittings within quantum dots introduce phase differences between the two biexciton decay paths that greatly reduce the entanglement of photon pairs generated via biexciton recombination. We analyze this problem in the frequency domain and propose a practicable method to compensate the phase difference by inserting a spatial light modulator, which substantially improves the entanglement of the photon pairs without any loss.Comment: 4 pages, 3 figure

Human B-cell Ontogeny in Humanized NOD/SCID γcnull^{null} Mice Generates a Diverse Yet Auto/Poly- and HIV-1 Reactive Antibody Repertoire

Characterization of the human antibody (Ab) repertoire in mouse models of the human immune system is essential to establish their relevance in translational studies. Single human B-cells were sorted from bone marrow and periphery of humanized NOD/SCID γc$^{null}$ mice at 8–10 months post-engraftment with human cord blood-derived CD34$^+$ stem cells. Human immunoglobulin variable heavy (V$_H$) and kappa (V$_κ$) genes were amplified, cognate V$_H$-V$_κ$ gene-pairs assembled as single-chain variable fragment-Fc antibodies (scFvFcs) and functional studies performed. Although overall distribution of V$_H$ genes approximated the normal human Ab repertoire, analysis of the V$_H$-third complementarity determining regions (H-CDR3) in the mature B-cell subset demonstrated an increase in length and positive charges suggesting autoimmune characteristics. Additionally, >70% of Vκ sequences utilized V$_κ$4-1, a germline gene associated with autoimmunity. The mature B-cell subset-derived scFvFcs displayed the highest frequency of autoreactivity and polyspecificity, suggesting defects in checkpoint control mechanisms. Furthermore, these scFvFcs demonstrated binding to recombinant HIV envelope corroborating previous observations of poly/autoreactivity in anti-HIVgp140 antibodies. These data lend support to the hypothesis that anti-HIV BnAbs may be derived from auto/polyspecific Abs that escaped immune elimination and that the hNSG mouse could provide a new experimental platform for studying the origin of anti-HIV neutralizing Ab responses