2,939 research outputs found
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 linear queries over a database , we
seek to find the differentially private mechanism that has the minimum mean
squared error. For pure differential privacy, an approximation to
the optimal mechanism is known. Our first contribution is to give an 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 . 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 and any given upper bound on . 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 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 new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)
We study the integer minimization of a quasiconvex polynomial with
quasiconvex polynomial constraints. We propose a new algorithm that is an
improvement upon the best known algorithm due to Heinz (Journal of Complexity,
2005). This improvement is achieved by applying a new modern Lenstra-type
algorithm, finding optimal ellipsoid roundings, and considering sparse
encodings of polynomials. For the bounded case, our algorithm attains a
time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound
on the number of monomials in each polynomial and r is the binary encoding
length of a bound on the feasible region. In the general case, s l^{O(1)}
d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total
degree of the polynomials and l bounds the maximum binary encoding size of the
input.Comment: 28 pages, 10 figure
Simple Approximations of Semialgebraic Sets and their Applications to Control
Many uncertainty sets encountered in control systems analysis and design can
be expressed in terms of semialgebraic sets, that is as the intersection of
sets described by means of polynomial inequalities. Important examples are for
instance the solution set of linear matrix inequalities or the Schur/Hurwitz
stability domains. These sets often have very complicated shapes (non-convex,
and even non-connected), which renders very difficult their manipulation. It is
therefore of considerable importance to find simple-enough approximations of
these sets, able to capture their main characteristics while maintaining a low
level of complexity. For these reasons, in the past years several convex
approximations, based for instance on hyperrect-angles, polytopes, or
ellipsoids have been proposed. In this work, we move a step further, and
propose possibly non-convex approximations , based on a small volume polynomial
superlevel set of a single positive polynomial of given degree. We show how
these sets can be easily approximated by minimizing the L1 norm of the
polynomial over the semialgebraic set, subject to positivity constraints.
Intuitively, this corresponds to the trace minimization heuristic commonly
encounter in minimum volume ellipsoid problems. From a computational viewpoint,
we design a hierarchy of linear matrix inequality problems to generate these
approximations, and we provide theoretically rigorous convergence results, in
the sense that the hierarchy of outer approximations converges in volume (or,
equivalently, almost everywhere and almost uniformly) to the original set. Two
main applications of the proposed approach are considered. The first one aims
at reconstruction/approximation of sets from a finite number of samples. In the
second one, we show how the concept of polynomial superlevel set can be used to
generate samples uniformly distributed on a given semialgebraic set. The
efficiency of the proposed approach is demonstrated by different numerical
examples
Collision-free path planning for robots using B-splines and simulated annealing
This thesis describes a technique to obtain an optimal collision-free path for an automated guided vehicle (AGV) and/or robot in two and three dimensions by synthesizing a B-spline curve under geometric and intrinsic constraints. The problem is formulated as a combinatorial optimization problem and solved by using simulated annealing. A two-link planar manipulator is included to show that the B-spline curve can also be synthesized by adding kinematic characteristics of the robot. A cost function, which includes obstacle proximity, excessive arc length, uneven parametric distribution and, possibly, link proximity costs, is developed for the simulated annealing algorithm. Three possible cases for the orientation of the moving object are explored: (a) fixed orientation, (b) orientation as another independent variable, and (c) orientation given by the slope of the curve. To demonstrate the robustness of the technique, several examples are presented. Objects are modeled as ellipsoid type shapes. The procedure to obtain the describing parameters of the ellipsoid is also presented
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