51 research outputs found
Logarithmic Representability of Integers as k-Sums
A set A=A_{k,n} in [n]\cup{0} is said to be an additive k-basis if each
element in {0,1,...,kn} can be written as a k-sum of elements of A in at least
one way. Seeking multiple representations as k-sums, and given any function
phi(n), with lim(phi(n))=infinity, we say that A is a truncated
phi(n)-representative k-basis for [n] if for each j in [alpha n, (k-alpha)n]
the number of ways that j can be represented as a k-sum of elements of A_{k,n}
is Theta(phi(n)). In this paper, we follow tradition and focus on the case
phi(n)=log n, and show that a randomly selected set in an appropriate
probability space is a truncated log-representative basis with probability that
tends to one as n tends to infinity. This result is a finite version of a
result proved by Erdos (1956) and extended by Erdos and Tetali (1990).Comment: 18 page
Sharp Threshold Asymptotics for the Emergence of Additive Bases
A subset A of {0,1,...,n} is said to be a 2-additive basis for {1,2,...,n} if
each j in {1,2,...,n} can be written as j=x+y, x,y in A, x<=y. If we pick each
integer in {0,1,...,n} independently with probability p=p_n tending to 0, thus
getting a random set A, what is the probability that we have obtained a
2-additive basis? We address this question when the target sum-set is
[(1-alpha)n,(1+alpha)n] (or equivalently [alpha n, (2-alpha) n]) for some
0<alpha<1. Under either model, the Stein-Chen method of Poisson approximation
is used, in conjunction with Janson's inequalities, to tease out a very sharp
threshold for the emergence of a 2-additive basis. Generalizations to
k-additive bases are then given.Comment: 22 page
Matched filters for noisy induced subgraph detection
First author draftWe consider the problem of finding the vertex correspondence between two graphs with different number of vertices where the smaller graph is still potentially large. We propose a solution to this problem via a graph matching matched filter: padding the smaller graph in different ways and then using graph matching methods to align it to the larger network. Under a statistical model for correlated pairs of graphs, which yields a noisy copy of the small graph within the larger graph, the resulting optimization problem can be guaranteed to recover the true vertex correspondence between the networks, though there are currently no efficient algorithms for solving this problem. We consider an approach that exploits a partially known correspondence and show via varied simulations and applications to the Drosophila connectome that in practice this approach can achieve good performance.https://arxiv.org/abs/1803.02423https://arxiv.org/abs/1803.0242
Matched Filters for Noisy Induced Subgraph Detection
The problem of finding the vertex correspondence between two noisy graphs
with different number of vertices where the smaller graph is still large has
many applications in social networks, neuroscience, and computer vision. We
propose a solution to this problem via a graph matching matched filter:
centering and padding the smaller adjacency matrix and applying graph matching
methods to align it to the larger network. The centering and padding schemes
can be incorporated into any algorithm that matches using adjacency matrices.
Under a statistical model for correlated pairs of graphs, which yields a noisy
copy of the small graph within the larger graph, the resulting optimization
problem can be guaranteed to recover the true vertex correspondence between the
networks.
However, there are currently no efficient algorithms for solving this
problem. To illustrate the possibilities and challenges of such problems, we
use an algorithm that can exploit a partially known correspondence and show via
varied simulations and applications to {\it Drosophila} and human connectomes
that this approach can achieve good performance.Comment: 41 pages, 7 figure
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