52,328 research outputs found
Differentially Private Release and Learning of Threshold Functions
We prove new upper and lower bounds on the sample complexity of differentially private algorithms for releasing approximate answers to
threshold functions. A threshold function over a totally ordered domain
evaluates to if , and evaluates to otherwise. We
give the first nontrivial lower bound for releasing thresholds with
differential privacy, showing that the task is impossible
over an infinite domain , and moreover requires sample complexity , which grows with the size of the domain. Inspired by the
techniques used to prove this lower bound, we give an algorithm for releasing
thresholds with samples. This improves the
previous best upper bound of (Beimel et al., RANDOM
'13).
Our sample complexity upper and lower bounds also apply to the tasks of
learning distributions with respect to Kolmogorov distance and of properly PAC
learning thresholds with differential privacy. The lower bound gives the first
separation between the sample complexity of properly learning a concept class
with differential privacy and learning without privacy. For
properly learning thresholds in dimensions, this lower bound extends to
.
To obtain our results, we give reductions in both directions from releasing
and properly learning thresholds and the simpler interior point problem. Given
a database of elements from , the interior point problem asks for an
element between the smallest and largest elements in . We introduce new
recursive constructions for bounding the sample complexity of the interior
point problem, as well as further reductions and techniques for proving
impossibility results for other basic problems in differential privacy.Comment: 43 page
Volume-Enclosing Surface Extraction
In this paper we present a new method, which allows for the construction of
triangular isosurfaces from three-dimensional data sets, such as 3D image data
and/or numerical simulation data that are based on regularly shaped, cubic
lattices. This novel volume-enclosing surface extraction technique, which has
been named VESTA, can produce up to six different results due to the nature of
the discretized 3D space under consideration. VESTA is neither template-based
nor it is necessarily required to operate on 2x2x2 voxel cell neighborhoods
only. The surface tiles are determined with a very fast and robust construction
technique while potential ambiguities are detected and resolved. Here, we
provide an in-depth comparison between VESTA and various versions of the
well-known and very popular Marching Cubes algorithm for the very first time.
In an application section, we demonstrate the extraction of VESTA isosurfaces
for various data sets ranging from computer tomographic scan data to simulation
data of relativistic hydrodynamic fireball expansions.Comment: 24 pages, 33 figures, 4 tables, final versio
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