The Geometry of Differential Privacy: the Sparse and Approximate Cases

In this work, we study trade-offs between accuracy and privacy in the context of linear queries over histograms. This is a rich class of queries that includes contingency tables and range queries, and has been a focus of a long line of work. For a set of $d$ linear queries over a database $x \in \R^N$, we seek to find the differentially private mechanism that has the minimum mean squared error. For pure differential privacy, an $O(\log^2 d)$ approximation to the optimal mechanism is known. Our first contribution is to give an $O(\log^2 d)$ approximation guarantee for the case of (\eps,\delta)-differential privacy. Our mechanism is simple, efficient and adds correlated Gaussian noise to the answers. We prove its approximation guarantee relative to the hereditary discrepancy lower bound of Muthukrishnan and Nikolov, using tools from convex geometry. We next consider this question in the case when the number of queries exceeds the number of individuals in the database, i.e. when $d > n \triangleq \|x\|_1$. It is known that better mechanisms exist in this setting. Our second main contribution is to give an (\eps,\delta)-differentially private mechanism which is optimal up to a \polylog(d,N) factor for any given query set $A$ and any given upper bound $n$ on $\|x\|_1$. This approximation is achieved by coupling the Gaussian noise addition approach with a linear regression step. We give an analogous result for the \eps-differential privacy setting. We also improve on the mean squared error upper bound for answering counting queries on a database of size $n$ by Blum, Ligett, and Roth, and match the lower bound implied by the work of Dinur and Nissim up to logarithmic factors. The connection between hereditary discrepancy and the privacy mechanism enables us to derive the first polylogarithmic approximation to the hereditary discrepancy of a matrix $A$

Renormalization group evolution of neutrino mixing parameters near θ13=0\theta_{13} = 0 and models with vanishing θ13\theta_{13} at the high scale

Renormalization group (RG) evolution of the neutrino mass matrix may take the value of the mixing angle $\theta_{13}$ very close to zero, or make it vanish. On the other hand, starting from $\theta_{13}=0$ at the high scale it may be possible to generate a non-zero $\theta_{13}$ radiatively. In the most general scenario with non-vanishing CP violating Dirac and Majorana phases, we explore the evolution in the vicinity of $\theta_{13}=0$, in terms of its structure in the complex ${\cal U}_{e3}$ plane. This allows us to explain the apparent singularity in the evolution of the Dirac CP phase $\delta$ at $\theta_{13}=0$. We also introduce a formalism for calculating the RG evolution of neutrino parameters that uses the Jarlskog invariant and naturally avoids this singular behaviour. We find that the parameters need to be extremely fine-tuned in order to get exactly vanishing $\theta_{13}$ during evolution. For the class of neutrino mass models with $\theta_{13}=0$ at the high scale, we calculate the extent to which RG evolution can generate a nonzero $\theta_{13}$, when the low energy effective theory is the standard model or its minimal supersymmetric extension. We find correlated constraints on $\theta_{13}$, the lightest neutrino mass $m_0$, the effective Majorana mass $m_{ee}$ measured in the neutrinoless double beta decay, and the supersymmetric parameter $\tan\beta$.Comment: 24 pages, 6 figures, revtex

Phenomenological implications of the Friedberg-Lee transformation in a neutrino mass model with μτ\mu\tau-flavored CP symmetry

We propose a neutrino mass model with $\mu\tau$-flavored CP symmetry, where the effective light neutrino Lagrangian enjoys an additional invariance under a Friedberg-Lee (FL) transformation on the left-handed flavor neutrino fields, that leads to a highly predictive and testable scenario. While both types of the light neutrino mass ordering, i.e., Normal Ordering (NO) as well as the Inverted Ordering (IO) are allowed, the absolute scale of neutrino masses is fixed by the vanishing determinant of light Majorana neutrino mass matrix $M_\nu$. We show that for both types of mass ordering, whilst the atmospheric mixing angle $\theta_{23}$ is in general nonmaximal ($\theta_{23}\neq \pi/4$), the Dirac CP phase $\delta$ is exactly maximal ($\delta=\pi/2,3\pi/2$) for IO and nearly maximal for NO owing to $\cos\delta\propto \sin\theta_{13}$. For the NO, very tiny nonvanishing Majorana CP violation might appear through one of the Majorana phases $\beta$; otherwise the model predicts vanishing Majorana CP violation. Thus, despite the fact, that from the measurement of $\theta_{23}$, it is difficult to rule out the model, any large deviation of $\delta$ from its maximality, will surely falsify the scenario. For a comprehensive numerical analysis, beside fitting the neutrino oscillation global fit data, we also present a study on the $\nu_\mu\rightarrow \nu_e$ oscillation which is expected to show up Dirac CP violation in different long baseline experiments. Finally, assuming purely astrophysical sources, we calculate the Ultra High Energy (UHE) neutrino flavor flux ratios at neutrino telescopes, such as IceCube, from which statements on the octant of $\theta_{23}$ could be made in our model.Comment: 20 pages, 7 figures, updated with clarifications and minor changes, version published in JHE

The Bivariate Normal Copula

We collect well known and less known facts about the bivariate normal distribution and translate them into copula language. In addition, we prove a very general formula for the bivariate normal copula, we compute Gini's gamma, and we provide improved bounds and approximations on the diagonal.Comment: 24 page

Leptogenesis and dark matter detection in a TeV scale neutrino mass model with inverted mass hierarchy

Realization of the inverted hierarchy is studied in the radiative neutrino mass model with an additional doublet, in which neutrino masses and dark matter could be induced from a common particle. We show that the sufficient baryon number asymmetry is generated through resonant leptogenesis even for the case with rather mild degeneracy among TeV scale right-handed neutrinos. We also discuss the relation between this neutrino mass generation mechanism and low energy experiments for the DM direct search, the neutrinoless double $\beta$ decay and so on.Comment: 25 pages, 5 figures, title changed, figures replaced and added, discussion on DM direct search extended, conclusions unchanged, published versio