16,117 research outputs found
Affine Buildings and Tropical Convexity
The notion of convexity in tropical geometry is closely related to notions of
convexity in the theory of affine buildings. We explore this relationship from
a combinatorial and computational perspective. Our results include a convex
hull algorithm for the Bruhat--Tits building of SL and techniques for
computing with apartments and membranes. While the original inspiration was the
work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel
and Tevelev in algebraic geometry, our tropical algorithms will also be
applicable to problems in other fields of mathematics.Comment: 22 pages, 4 figure
Robust Classification for Imprecise Environments
In real-world environments it usually is difficult to specify target
operating conditions precisely, for example, target misclassification costs.
This uncertainty makes building robust classification systems problematic. We
show that it is possible to build a hybrid classifier that will perform at
least as well as the best available classifier for any target conditions. In
some cases, the performance of the hybrid actually can surpass that of the best
known classifier. This robust performance extends across a wide variety of
comparison frameworks, including the optimization of metrics such as accuracy,
expected cost, lift, precision, recall, and workforce utilization. The hybrid
also is efficient to build, to store, and to update. The hybrid is based on a
method for the comparison of classifier performance that is robust to imprecise
class distributions and misclassification costs. The ROC convex hull (ROCCH)
method combines techniques from ROC analysis, decision analysis and
computational geometry, and adapts them to the particulars of analyzing learned
classifiers. The method is efficient and incremental, minimizes the management
of classifier performance data, and allows for clear visual comparisons and
sensitivity analyses. Finally, we point to empirical evidence that a robust
hybrid classifier indeed is needed for many real-world problems.Comment: 24 pages, 12 figures. To be published in Machine Learning Journal.
For related papers, see http://www.hpl.hp.com/personal/Tom_Fawcett/ROCCH
The convex hull of a finite set
We study -separately convex hulls of finite
sets of points in , as introduced in
\cite{KirchheimMullerSverak2003}. When is considered as a
certain subset of matrices, this notion of convexity corresponds
to rank-one convex convexity . If is identified instead
with a subset of matrices, it actually agrees with the quasiconvex
hull, due to a recent result \cite{HarrisKirchheimLin18}.
We introduce " complexes", which generalize constructions. For a
finite set , a " -complex" is a complex whose extremal points
belong to . The "-complex convex hull of ", , is the union
of all -complexes. We prove that is contained in the
convex hull .
We also consider outer approximations to convexity based in the
locality theorem \cite[4.7]{Kirchheim2003}. Starting with a crude outer
approximation we iteratively chop off "-prisms". For the examples in
\cite{KirchheimMullerSverak2003}, and many others, this procedure reaches a
" -complex" in a finite number of steps, and thus computes the
convex hull.
We show examples of finite sets for which this procedure does not reach the
convex hull in finite time, but we show that a sequence of outer
approximations built with -prisms converges to a -complex. We
conclude that is always a " -complex", which has interesting
consequences
Topomap: Topological Mapping and Navigation Based on Visual SLAM Maps
Visual robot navigation within large-scale, semi-structured environments
deals with various challenges such as computation intensive path planning
algorithms or insufficient knowledge about traversable spaces. Moreover, many
state-of-the-art navigation approaches only operate locally instead of gaining
a more conceptual understanding of the planning objective. This limits the
complexity of tasks a robot can accomplish and makes it harder to deal with
uncertainties that are present in the context of real-time robotics
applications. In this work, we present Topomap, a framework which simplifies
the navigation task by providing a map to the robot which is tailored for path
planning use. This novel approach transforms a sparse feature-based map from a
visual Simultaneous Localization And Mapping (SLAM) system into a
three-dimensional topological map. This is done in two steps. First, we extract
occupancy information directly from the noisy sparse point cloud. Then, we
create a set of convex free-space clusters, which are the vertices of the
topological map. We show that this representation improves the efficiency of
global planning, and we provide a complete derivation of our algorithm.
Planning experiments on real world datasets demonstrate that we achieve similar
performance as RRT* with significantly lower computation times and storage
requirements. Finally, we test our algorithm on a mobile robotic platform to
prove its advantages.Comment: 8 page
Convexity-Increasing Morphs of Planar Graphs
We study the problem of convexifying drawings of planar graphs. Given any
planar straight-line drawing of an internally 3-connected graph, we show how to
morph the drawing to one with strictly convex faces while maintaining planarity
at all times. Our morph is convexity-increasing, meaning that once an angle is
convex, it remains convex. We give an efficient algorithm that constructs such
a morph as a composition of a linear number of steps where each step either
moves vertices along horizontal lines or moves vertices along vertical lines.
Moreover, we show that a linear number of steps is worst-case optimal.
To obtain our result, we use a well-known technique by Hong and Nagamochi for
finding redrawings with convex faces while preserving y-coordinates. Using a
variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and
Nagamochi's result which comes with a better running time. This is of
independent interest, as Hong and Nagamochi's technique serves as a building
block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201
Convex Hull Formation for Programmable Matter
We envision programmable matter as a system of nano-scale agents (called
particles) with very limited computational capabilities that move and compute
collectively to achieve a desired goal. We use the geometric amoebot model as
our computational framework, which assumes particles move on the triangular
lattice. Motivated by the problem of sealing an object using minimal resources,
we show how a particle system can self-organize to form an object's convex
hull. We give a distributed, local algorithm for convex hull formation and
prove that it runs in asynchronous rounds, where is the
length of the object's boundary. Within the same asymptotic runtime, this
algorithm can be extended to also form the object's (weak) -hull,
which uses the same number of particles but minimizes the area enclosed by the
hull. Our algorithms are the first to compute convex hulls with distributed
entities that have strictly local sensing, constant-size memory, and no shared
sense of orientation or coordinates. Ours is also the first distributed
approach to computing restricted-orientation convex hulls. This approach
involves coordinating particles as distributed memory; thus, as a supporting
but independent result, we present and analyze an algorithm for organizing
particles with constant-size memory as distributed binary counters that
efficiently support increments, decrements, and zero-tests --- even as the
particles move
The Complexity of Order Type Isomorphism
The order type of a point set in maps each -tuple of points to
its orientation (e.g., clockwise or counterclockwise in ). Two point sets
and have the same order type if there exists a mapping from to
for which every -tuple of and the
corresponding tuple in have the same
orientation. In this paper we investigate the complexity of determining whether
two point sets have the same order type. We provide an algorithm for
this task, thereby improving upon the algorithm
of Goodman and Pollack (1983). The algorithm uses only order type queries and
also works for abstract order types (or acyclic oriented matroids). Our
algorithm is optimal, both in the abstract setting and for realizable points
sets if the algorithm only uses order type queries.Comment: Preliminary version of paper to appear at ACM-SIAM Symposium on
Discrete Algorithms (SODA14
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