5,258 research outputs found
Subsampling in Smoothed Range Spaces
We consider smoothed versions of geometric range spaces, so an element of the
ground set (e.g. a point) can be contained in a range with a non-binary value
in . Similar notions have been considered for kernels; we extend them to
more general types of ranges. We then consider approximations of these range
spaces through -nets and -samples (aka
-approximations). We characterize when size bounds for
-samples on kernels can be extended to these more general
smoothed range spaces. We also describe new generalizations for -nets to these range spaces and show when results from binary range spaces can
carry over to these smoothed ones.Comment: This is the full version of the paper which appeared in ALT 2015. 16
pages, 3 figures. In Algorithmic Learning Theory, pp. 224-238. Springer
International Publishing, 201
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
Periodic Planar Disk Packings
Several conditions are given when a packing of equal disks in a torus is
locally maximally dense, where the torus is defined as the quotient of the
plane by a two-dimensional lattice. Conjectures are presented that claim that
the density of any strictly jammed packings, whose graph does not consist of
all triangles and the torus lattice is the standard triangular lattice, is at
most , where is the number of packing
disks. Several classes of collectively jammed packings are presented where the
conjecture holds.Comment: 26 pages, 13 figure
Finite-sample Analysis of M-estimators using Self-concordance
We demonstrate how self-concordance of the loss can be exploited to obtain
asymptotically optimal rates for M-estimators in finite-sample regimes. We
consider two classes of losses: (i) canonically self-concordant losses in the
sense of Nesterov and Nemirovski (1994), i.e., with the third derivative
bounded with the power of the second; (ii) pseudo self-concordant losses,
for which the power is removed, as introduced by Bach (2010). These classes
contain some losses arising in generalized linear models, including logistic
regression; in addition, the second class includes some common pseudo-Huber
losses. Our results consist in establishing the critical sample size sufficient
to reach the asymptotically optimal excess risk for both classes of losses.
Denoting the parameter dimension, and the effective
dimension which takes into account possible model misspecification, we find the
critical sample size to be for canonically
self-concordant losses, and for pseudo
self-concordant losses, where is the problem-dependent local curvature
parameter. In contrast to the existing results, we only impose local
assumptions on the data distribution, assuming that the calibrated design,
i.e., the design scaled with the square root of the second derivative of the
loss, is subgaussian at the best predictor . Moreover, we obtain the
improved bounds on the critical sample size, scaling near-linearly in
, under the extra assumption that the calibrated design
is subgaussian in the Dikin ellipsoid of . Motivated by these
findings, we construct canonically self-concordant analogues of the Huber and
logistic losses with improved statistical properties. Finally, we extend some
of these results to -regularized M-estimators in high dimensions
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Experimental Realization of an Extreme-Parameter Omnidirectional Cloak
An ideal transformation-based omnidirectional cloak always relies on metamaterials with extreme parameters, which were previously thought to be too difficult to realize. For such a reason, in previous experimental proposals of invisibility cloaks, the extreme parameters requirements are usually abandoned, leading to inherent scattering. Here, we report on the first experimental demonstration of an omnidirectional cloak that satisfies the extreme parameters requirement, which can hide objects in a homogenous background. Instead of using resonant metamaterials that usually involve unavoidable absorptive loss, the extreme parameters are achieved using a nonresonant metamaterial comprising arrays of subwavelength metallic channels manufactured with 3D metal printing technology. A high level transmission of electromagnetic wave propagating through the present omnidirectional cloak, as well as significant reduction of scattering field, is demonstrated both numerically and experimentally. Our work may also inspire experimental realizations of the other full-parameter omnidirectional optical devices such as concentrator, rotators, and optical illusion apparatuses
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