4,645 research outputs found
Computational Methods for Sparse Solution of Linear Inverse Problems
The goal of the sparse approximation problem is to approximate a target signal using a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a plethora of applications
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
Average case polyhedral complexity of the maximum stable set problem
We study the minimum number of constraints needed to formulate random
instances of the maximum stable set problem via linear programs (LPs), in two
distinct models. In the uniform model, the constraints of the LP are not
allowed to depend on the input graph, which should be encoded solely in the
objective function. There we prove a lower bound with
probability at least for every LP that is exact for a randomly
selected set of instances; each graph on at most n vertices being selected
independently with probability . In the
non-uniform model, the constraints of the LP may depend on the input graph, but
we allow weights on the vertices. The input graph is sampled according to the
G(n, p) model. There we obtain upper and lower bounds holding with high
probability for various ranges of p. We obtain a super-polynomial lower bound
all the way from to . Our upper bound is close to this as there is only an essentially quadratic
gap in the exponent, which currently also exists in the worst-case model.
Finally, we state a conjecture that would close this gap, both in the
average-case and worst-case models
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
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