6,292 research outputs found
Regression Depth and Center Points
We show that, for any set of n points in d dimensions, there exists a
hyperplane with regression depth at least ceiling(n/(d+1)). as had been
conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n
hyperplanes in d dimensions there exists a point that cannot escape to infinity
without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our
approach to related questions on the existence of partitions of the data into
subsets such that a common plane has nonzero regression depth in each subset,
and to the computational complexity of regression depth problems.Comment: 14 pages, 3 figure
Applications of incidence bounds in point covering problems
In the Line Cover problem a set of n points is given and the task is to cover
the points using either the minimum number of lines or at most k lines. In
Curve Cover, a generalization of Line Cover, the task is to cover the points
using curves with d degrees of freedom. Another generalization is the
Hyperplane Cover problem where points in d-dimensional space are to be covered
by hyperplanes. All these problems have kernels of polynomial size, where the
parameter is the minimum number of lines, curves, or hyperplanes needed. First
we give a non-parameterized algorithm for both problems in O*(2^n) (where the
O*(.) notation hides polynomial factors of n) time and polynomial space,
beating a previous exponential-space result. Combining this with incidence
bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover
algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant.
Our result improves the previous best times O*((k/1.35)^k) for Line Cover
(where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds
for covering points by parabolas or conics. We also present an algorithm for
Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving
on the previous time of O*((k^2/1.3)^k).Comment: SoCG 201
The parameterized complexity of some geometric problems in unbounded dimension
We study the parameterized complexity of the following fundamental geometric
problems with respect to the dimension : i) Given points in \Rd,
compute their minimum enclosing cylinder. ii) Given two -point sets in
\Rd, decide whether they can be separated by two hyperplanes. iii) Given a
system of linear inequalities with variables, find a maximum-size
feasible subsystem. We show that (the decision versions of) all these problems
are W[1]-hard when parameterized by the dimension . %and hence not solvable
in time, for any computable function and constant
%(unless FPT=W[1]). Our reductions also give a -time lower bound
(under the Exponential Time Hypothesis)
Fast computation of Tukey trimmed regions and median in dimension
Given data in , a Tukey -trimmed region is the set of
all points that have at least Tukey depth w.r.t. the data. As they are
visual, affine equivariant and robust, Tukey regions are useful tools in
nonparametric multivariate analysis. While these regions are easily defined and
interpreted, their practical use in applications has been impeded so far by the
lack of efficient computational procedures in dimension . We construct
two novel algorithms to compute a Tukey -trimmed region, a na\"{i}ve
one and a more sophisticated one that is much faster than known algorithms.
Further, a strict bound on the number of facets of a Tukey region is derived.
In a large simulation study the novel fast algorithm is compared with the
na\"{i}ve one, which is slower and by construction exact, yielding in every
case the same correct results. Finally, the approach is extended to an
algorithm that calculates the innermost Tukey region and its barycenter, the
Tukey median
Convex Hulls under Uncertainty
We study the convex-hull problem in a probabilistic setting, motivated by the
need to handle data uncertainty inherent in many applications, including sensor
databases, location-based services and computer vision. In our framework, the
uncertainty of each input site is described by a probability distribution over
a finite number of possible locations including a \emph{null} location to
account for non-existence of the point. Our results include both exact and
approximation algorithms for computing the probability of a query point lying
inside the convex hull of the input, time-space tradeoffs for the membership
queries, a connection between Tukey depth and membership queries, as well as a
new notion of \some-hull that may be a useful representation of uncertain
hulls
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