181,702 research outputs found
Long Dominating Cycles in Graphs
All graphs considered in this paper will be finite and simple. We use Bondy & Murty for terminology and notations not defined here
Rates for branching particle approximations of continuous-discrete filters
Herein, we analyze an efficient branching particle method for asymptotic
solutions to a class of continuous-discrete filtering problems. Suppose that
is a Markov process and we wish to calculate the measure-valued
process , where and is a distorted, corrupted, partial
observation of . Then, one constructs a particle system with
observation-dependent branching and initial particles whose empirical
measure at time , , closely approximates . Each particle
evolves independently of the other particles according to the law of the signal
between observation times , and branches with small probability at an
observation time. For filtering problems where is very small, using
the algorithm considered in this paper requires far fewer computations than
other algorithms that branch or interact all particles regardless of the value
of . We analyze the algorithm on L\'{e}vy-stable signals and give
rates of convergence for , where
is a Sobolev norm, as well as related convergence
results.Comment: Published at http://dx.doi.org/10.1214/105051605000000539 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
The Capacity of Private Computation
We introduce the problem of private computation, comprised of distributed
and non-colluding servers, independent datasets, and a user who wants to
compute a function of the datasets privately, i.e., without revealing which
function he wants to compute, to any individual server. This private
computation problem is a strict generalization of the private information
retrieval (PIR) problem, obtained by expanding the PIR message set (which
consists of only independent messages) to also include functions of those
messages. The capacity of private computation, , is defined as the maximum
number of bits of the desired function that can be retrieved per bit of total
download from all servers. We characterize the capacity of private computation,
for servers and independent datasets that are replicated at each
server, when the functions to be computed are arbitrary linear combinations of
the datasets. Surprisingly, the capacity,
, matches the capacity of PIR with
servers and messages. Thus, allowing arbitrary linear computations does
not reduce the communication rate compared to pure dataset retrieval. The same
insight is shown to hold even for arbitrary non-linear computations when the
number of datasets
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