Differential Privacy in Metric Spaces: Numerical, Categorical and Functional Data Under the One Roof

We study Differential Privacy in the abstract setting of Probability on metric spaces. Numerical, categorical and functional data can be handled in a uniform manner in this setting. We demonstrate how mechanisms based on data sanitisation and those that rely on adding noise to query responses fit within this framework. We prove that once the sanitisation is differentially private, then so is the query response for any query. We show how to construct sanitisations for high-dimensional databases using simple 1-dimensional mechanisms. We also provide lower bounds on the expected error for differentially private sanitisations in the general metric space setting. Finally, we consider the question of sufficient sets for differential privacy and show that for relaxed differential privacy, any algebra generating the Borel $\sigma$-algebra is a sufficient set for relaxed differential privacy.Comment: 18 Page

Period preserving nonisospectral flows and the moduli space of periodic solutions of soliton equations

Flows on the moduli space of the algebraic Riemann surfaces, preserving the periods of the corresponding solutions of the soliton equations are studied. We show that these flows are gradient with respect to some indefinite symmetric flat metric arising in the Hamiltonian theory of the Whitham equations. The functions generating these flows are conserved quantities for all the equations simultaneously. We show that for 1+1 systems these flows can be imbedded in a larger system of ordinary nonlinear differential equations with a rational right-hand side. Finally these flows are used to give a complete description of the moduli space of algebraic Riemann surfaces corresponding to periodic solutions of the nonlinear Schr\"odinger equation.Comment: 35 pages, LaTex. Macros file elsart.sty is used (it was submitted by the authors to [email protected] library macroses),e-mail: [email protected], e-mail:[email protected]

QUASII: QUery-Aware Spatial Incremental Index.

With large-scale simulations of increasingly detailed models and improvement of data acquisition technologies, massive amounts of data are easily and quickly created and collected. Traditional systems require indexes to be built before analytic queries can be executed efficiently. Such an indexing step requires substantial computing resources and introduces a considerable and growing data-to-insight gap where scientists need to wait before they can perform any analysis. Moreover, scientists often only use a small fraction of the data - the parts containing interesting phenomena - and indexing it fully does not always pay off. In this paper we develop a novel incremental index for the exploration of spatial data. Our approach, QUASII, builds a data-oriented index as a side-effect of query execution. QUASII distributes the cost of indexing across all queries, while building the index structure only for the subset of data queried. It reduces data-to-insight time and curbs the cost of incremental indexing by gradually and partially sorting the data, while producing a data-oriented hierarchical structure at the same time. As our experiments show, QUASII reduces the data-to-insight time by up to a factor of 11.4x, while its performance converges to that of the state-of-the-art static indexes