6,912 research outputs found
Heavy Hitters and the Structure of Local Privacy
We present a new locally differentially private algorithm for the heavy
hitters problem which achieves optimal worst-case error as a function of all
standardly considered parameters. Prior work obtained error rates which depend
optimally on the number of users, the size of the domain, and the privacy
parameter, but depend sub-optimally on the failure probability.
We strengthen existing lower bounds on the error to incorporate the failure
probability, and show that our new upper bound is tight with respect to this
parameter as well. Our lower bound is based on a new understanding of the
structure of locally private protocols. We further develop these ideas to
obtain the following general results beyond heavy hitters.
Advanced Grouposition: In the local model, group privacy for
users degrades proportionally to , instead of linearly in
as in the central model. Stronger group privacy yields improved max-information
guarantees, as well as stronger lower bounds (via "packing arguments"), over
the central model.
Building on a transformation of Bassily and Smith (STOC 2015), we
give a generic transformation from any non-interactive approximate-private
local protocol into a pure-private local protocol. Again in contrast with the
central model, this shows that we cannot obtain more accurate algorithms by
moving from pure to approximate local privacy
Exact Random Coding Secrecy Exponents for the Wiretap Channel
We analyze the exact exponential decay rate of the expected amount of
information leaked to the wiretapper in Wyner's wiretap channel setting using
wiretap channel codes constructed from both i.i.d. and constant-composition
random codes. Our analysis for those sampled from i.i.d. random coding ensemble
shows that the previously-known achievable secrecy exponent using this ensemble
is indeed the exact exponent for an average code in the ensemble. Furthermore,
our analysis on wiretap channel codes constructed from the ensemble of
constant-composition random codes leads to an exponent which, in addition to
being the exact exponent for an average code, is larger than the achievable
secrecy exponent that has been established so far in the literature for this
ensemble (which in turn was known to be smaller than that achievable by wiretap
channel codes sampled from i.i.d. random coding ensemble). We show examples
where the exact secrecy exponent for the wiretap channel codes constructed from
random constant-composition codes is larger than that of those constructed from
i.i.d. random codes and examples where the exact secrecy exponent for the
wiretap channel codes constructed from i.i.d. random codes is larger than that
of those constructed from constant-composition random codes. We, hence,
conclude that, unlike the error correction problem, there is no general
ordering between the two random coding ensembles in terms of their secrecy
exponent.Comment: 23 pages, 5 figures, submitted to IEEE Transactions on Information
Theor
Designing structured tight frames via an alternating projection method
Tight frames, also known as general Welch-bound- equality sequences, generalize orthonormal systems. Numerous applications - including communications, coding, and sparse approximation- require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm
Grid generation for the solution of partial differential equations
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given
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