200 research outputs found
The Complexity of Recognizing Geometric Hypergraphs
As set systems, hypergraphs are omnipresent and have various representations
ranging from Euler and Venn diagrams to contact representations. In a geometric
representation of a hypergraph , each vertex is associated
with a point and each hyperedge is associated
with a connected set such that for all . We say that a given
hypergraph is representable by some (infinite) family of sets in
, if there exist and such
that is a geometric representation of . For a family F, we define
RECOGNITION(F) as the problem to determine if a given hypergraph is
representable by F. It is known that the RECOGNITION problem is
-hard for halfspaces in . We study the
families of translates of balls and ellipsoids in , as well as of
other convex sets, and show that their RECOGNITION problems are also
-complete. This means that these recognition problems are
equivalent to deciding whether a multivariate system of polynomial equations
with integer coefficients has a real solution.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023) 17 pages, 11 figure
On embeddings of CAT(0) cube complexes into products of trees
We prove that the contact graph of a 2-dimensional CAT(0) cube complex of maximum degree can be coloured with at most
colours, for a fixed constant . This implies
that (and the associated median graph) isometrically embeds in the
Cartesian product of at most trees, and that the event
structure whose domain is admits a nice labeling with
labels. On the other hand, we present an example of a
5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes
which cannot be embedded into a Cartesian product of a finite number of trees.
This answers in the negative a question raised independently by F. Haglund, G.
Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the
computation of the bounds in Theorem 1. Some figures repaire
Cocompactly cubulated crystallographic groups
We prove that the simplicial boundary of a CAT(0) cube complex admitting a
proper, cocompact action by a virtually \integers^n group is isomorphic to
the hyperoctahedral triangulation of , providing a class of groups
for which the simplicial boundary of a -cocompact cube complex depends only
on . We also use this result to show that the cocompactly cubulated
crystallographic groups in dimension are precisely those that are
\emph{hyperoctahedral}. We apply this result to answer a question of Wise on
cocompactly cubulating virtually free abelian groups.Comment: Several correction
Average-case Complexity of Teaching Convex Polytopes via Halfspace Queries
We examine the task of locating a target region among those induced by intersections of n halfspaces in R^d. This generic task connects to fundamental machine learning problems, such as training a perceptron and learning a ϕ-separable dichotomy. We investigate the average teaching complexity of the task, i.e., the minimal number of samples (halfspace queries) required by a teacher to help a version-space learner in locating a randomly selected target. As our main result, we show that the average-case teaching complexity is Θ(d), which is in sharp contrast to the worst-case teaching complexity of Θ(n). If instead, we consider the average-case learning complexity, the bounds have a dependency on n as Θ(n) for i.i.d. queries and Θ(dlog(n)) for actively chosen queries by the learner. Our proof techniques are based on novel insights from computational geometry, which allow us to count the number of convex polytopes and faces in a Euclidean space depending on the arrangement of halfspaces. Our insights allow us to establish a tight bound on the average-case complexity for ϕ-separable dichotomies, which generalizes the known O(d) bound on the average number of "extreme patterns" in the classical computational geometry literature (Cover, 1965)
Inquisitive Pattern Recognition
The Department of Defense and the Department of the Air Force have funded automatic target recognition for several decades with varied success. The foundation of automatic target recognition is based upon pattern recognition. In this work, we present new pattern recognition concepts specifically in the area of classification and propose new techniques that will allow one to determine when a classifier is being arrogant. Clearly arrogance in classification is an undesirable attribute. A human is being arrogant when their expressed conviction in a decision overstates their actual experience in making similar decisions. Likewise given an input feature vector, we say a classifier is arrogant in its classification if its veracity is high yet its experience is low. Conversely a classifier is non-arrogant in its classification if there is a reasonable balance between its veracity and its experience. We quantify this balance and we discuss new techniques that will detect arrogance in a classifier. Inquisitiveness is in many ways the opposite of arrogance. In nature inquisitiveness is an eagerness for knowledge characterized by the drive to question to seek a deeper understanding and to challenge assumptions. The human capacity to doubt present beliefs allows us to acquire new experiences and to learn from our mistakes. Within the discrete world of computers, inquisitive pattern recognition is the constructive investigation and exploitation of conflict in information. This research defines inquisitiveness within the context of self-supervised machine learning and introduces mathematical theory and computational methods for quantifying incompleteness that is for isolating unstable, nonrepresentational regions in present information models
Medians in Median Graphs and Their Cube Complexes in Linear Time
The median of a set of vertices P of a graph G is the set of all vertices x of G minimizing the sum of distances from x to all vertices of P. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the ??-cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure. Our algorithm is based on the majority rule characterization of medians in median graphs and on a fast computation of parallelism classes of edges (?-classes or hyperplanes) via Lexicographic Breadth First Search (LexBFS). To prove the correctness of our algorithm, we show that any LexBFS ordering of the vertices of G satisfies the following fellow traveler property of independent interest: the parents of any two adjacent vertices of G are also adjacent
Sign rank versus VC dimension
This work studies the maximum possible sign rank of sign
matrices with a given VC dimension . For , this maximum is {three}. For
, this maximum is . For , similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an multiplicative
approximation for the sign rank.
We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the adjacency
matrix of a regular graph with a second eigenvalue of absolute value
and . We show that the sign rank of the signed
version of this matrix is at least . We use this connection to
prove the existence of a maximum class with VC
dimension and sign rank . This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran
Medians in median graphs and their cube complexes in linear time
The median of a set of vertices of a graph is the set of all vertices
of minimizing the sum of distances from to all vertices of . In
this paper, we present a linear time algorithm to compute medians in median
graphs, improving over the existing quadratic time algorithm. We also present a
linear time algorithm to compute medians in the -cube complexes
associated with median graphs. Median graphs constitute the principal class of
graphs investigated in metric graph theory and have a rich geometric and
combinatorial structure, due to their bijections with CAT(0) cube complexes and
domains of event structures. Our algorithm is based on the majority rule
characterization of medians in median graphs and on a fast computation of
parallelism classes of edges (-classes or hyperplanes) via
Lexicographic Breadth First Search (LexBFS). To prove the correctness of our
algorithm, we show that any LexBFS ordering of the vertices of satisfies
the following fellow traveler property of independent interest: the parents of
any two adjacent vertices of are also adjacent. Using the fast computation
of the -classes, we also compute the Wiener index (total distance) of
in linear time and the distance matrix in optimal quadratic time
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