Results 1  10
of
36
Clustering for EdgeCost Minimization
"... Leonard J. Schulman College of Computing Georgia Institute of Technology Atlanta GA 303320280 ABSTRACT We address the problem of partitioning a set of n points into clusters, so as to minimize the sum, over all intracluster pairs of points, of the cost associated with each pair. We obtain a ra ..."
Abstract

Cited by 32 (5 self)
 Add to MetaCart
Leonard J. Schulman College of Computing Georgia Institute of Technology Atlanta GA 303320280 ABSTRACT We address the problem of partitioning a set of n points into clusters, so as to minimize the sum, over all intracluster pairs of points, of the cost associated with each pair. We obtain a randomized approximation algorithm for this problem, for the cost functions ` 2 2 ; `1 and `2 , as well as any cost function isometrically embeddable in ` 2 2 .
On the Complexity of Computing and Learning with Multiplicative Neural Networks
 NEURAL COMPUTATION
"... In a great variety of neuron models neural inputs are combined using the summing operation. We introduce the concept of multiplicative neural networks that contain units which multiply their inputs instead of summing them and, thus, allow inputs to interact nonlinearly. The class of multiplicative n ..."
Abstract

Cited by 24 (3 self)
 Add to MetaCart
In a great variety of neuron models neural inputs are combined using the summing operation. We introduce the concept of multiplicative neural networks that contain units which multiply their inputs instead of summing them and, thus, allow inputs to interact nonlinearly. The class of multiplicative neural networks comprises such widely known and well studied network types as higherorder networks and product unit networks. We investigate the complexity of computing and learning for multiplicative neural networks. In particular, we derive upper and lower bounds on the VapnikChervonenkis (VC) dimension and the pseudo dimension for various types of networks with multiplicative units. As the most general case, we consider feedforward networks consisting of product and sigmoidal units, showing that their pseudo dimension is bounded from above by a polynomial with the same order of magnitude as the currently best known bound for purely sigmoidal networks. Moreover, we show that this bound holds even in the case when the unit type, product or sigmoidal, may be learned. Crucial for these results are calculations of solution set components bounds for new network classes. As to lower bounds we construct product unit networks of fixed depth with superlinear VC dimension. For sigmoidal networks of higher order we establish polynomial bounds that, in contrast to previous results, do not involve any restriction of the network order. We further consider various classes of higherorder units, also known as sigmapi units, that are characterized by connectivity constraints. In terms of these we derive some asymptotically tight bounds.
Randomized Approximation Algorithms for Set Multicover Problems with Applications to Reverse Engineering of Protein and Gene Networks
 Proc. 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX
, 2004
"... In this paper we investigate the computational complexities of a combinatorial problem that arises in the reverse engineering of protein and gene networks. Our contributions are as follows: . We abstract a combinatorial version of the problem and observe that this is "equivalent" to the set multi ..."
Abstract

Cited by 17 (11 self)
 Add to MetaCart
In this paper we investigate the computational complexities of a combinatorial problem that arises in the reverse engineering of protein and gene networks. Our contributions are as follows: . We abstract a combinatorial version of the problem and observe that this is "equivalent" to the set multicover problem when the "coverage" factor k is a function of the number of elements n of the universe. An important special case for our application is the case in which k = n  1.
Variational Principles for Circle Patterns
, 2003
"... A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given interse ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the circles, whether it is unique and how it may be constructed. Somewhat more generally, we allow conelike singularities in the centers and intersection points of the circles. We prove existence and uniqueness theorems for the solution of the circle pattern problem using a variational principle. The functionals (one for the euclidean, one for the hyperbolic case) are convex functions of the radii of the circles. The critical points correspond to solutions of the circle pattern problem. The analogous functional for the spherical case is not convex, hence this case is treated by stereographic projection to the plane. From the existence and uniqueness of circle patterns in the sphere, we derive a strengthened version of Steinitz’ theorem on the geometric realizability of abstract polyhedra. We derive the
A graph theoretical GaussBonnetChern theorem
, 2011
"... Abstract. We prove a discrete GaussBonnetChern theorem ∑ g∈V K(g) = χ(G) for finite graphs G = (V,E), where V is the vertex set and E is the edge set of the graph. The dimension of the graph, the local curvature form K and the Euler characteristic are all defined graph theoretically. ..."
Abstract

Cited by 12 (12 self)
 Add to MetaCart
Abstract. We prove a discrete GaussBonnetChern theorem ∑ g∈V K(g) = χ(G) for finite graphs G = (V,E), where V is the vertex set and E is the edge set of the graph. The dimension of the graph, the local curvature form K and the Euler characteristic are all defined graph theoretically.
Small examples of nonconstructible simplicial balls and spheres
 SIAM J. Discrete Math
, 2004
"... We construct nonconstructible simplicial dspheres with d + 10 vertices and nonconstructible, nonrealizable simplicial dballs with d + 9 vertices for d≥3. 1 ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
We construct nonconstructible simplicial dspheres with d + 10 vertices and nonconstructible, nonrealizable simplicial dballs with d + 9 vertices for d≥3. 1
Face Numbers of 4Polytopes and 3Spheres
 Proceedings of the international congress of mathematicians, ICM 2002
, 2002
"... Steinitz (1906) gave a remarkably simple and explicit description of the set of all fvectors f(P ) = (f0 , f1 , f2) of all 3dimensional convex polytopes. His result also identifies the simple and the simplicial 3dimensional polytopes as the only extreme cases. Moreover, it can be extended to s ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
Steinitz (1906) gave a remarkably simple and explicit description of the set of all fvectors f(P ) = (f0 , f1 , f2) of all 3dimensional convex polytopes. His result also identifies the simple and the simplicial 3dimensional polytopes as the only extreme cases. Moreover, it can be extended to strongly regular CW 2spheres (topological objects), and further to Eulerian lattices of length 4 (combinatorial objects).
A VARIATIONAL PRINCIPLE FOR WEIGHTED DELAUNAY TRIANGULATIONS AND HYPERIDEAL
"... We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with nonintersecting sitecircles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperb ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with nonintersecting sitecircles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus, the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, we consider weighted Delaunay triangulations in piecewise flat surfaces, allowing cone singularities with prescribed cone angles in the vertices. The material presented here extends work by Rivin on Delaunay triangulations and ideal polyhedra. 1.
Algorithmic issues in reverse engineering of protein and gene networks via the modular response analysis method, to appear in Annals of the New York Academy of Sciences (volume title: Reverse Engineering Biological Networks: Opportunities and Challenges i
 Annals of the NY Academy of Sciences
, 2007
"... This paper studies a computational problem motivated by the modular response analysis method for reverse engineering of protein and gene networks. This setcover problem is hard to solve exactly for large networks, but efficient approximation algorithms are given and their complexity is analyzed. ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
This paper studies a computational problem motivated by the modular response analysis method for reverse engineering of protein and gene networks. This setcover problem is hard to solve exactly for large networks, but efficient approximation algorithms are given and their complexity is analyzed.
An index formula for simple graphs
, 2012
"... Abstract. We prove that any odd dimensional geometric graph G = (V, E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 [1 − 2 χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
Abstract. We prove that any odd dimensional geometric graph G = (V, E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 [1 − 2 χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is a discrete analogue of the index of the gradient vector field ∇f and where Bf (x) is a graph defined by G and f. The PoincaréHopf formula ∑ x jf (x) = χ(G) allows so to express the Euler characteristic χ(G) of G in terms of smaller dimensional graphs defined by the unit sphere S(x) and the ”hypersurface graphs ” Bf (x). For odd dimensional geometric graphs, Bf (x) is a geometric graph of dimension dim(G)−2 and jf (x) = −χ(Bf (x))/2 = 0 implying χ(G) = 0 and zero curvature K(x) = 0 for all x. For even dimensional geometric graphs, the formula becomes jf (x) = 1−χ(Bf (x))/2 and allows with PoincaréHopf to write the Euler characteristic of G as a sum of the Euler characteristic of smaller dimensional graphs. The same integral geometric index formula also is valid for compact Riemannian manifolds M if f is a Morse function, S(x) is a sufficiently small geodesic sphere around x and Bf (x) = S(x) ∩ {y  f(y) = f(x)}. 1.