1,082 research outputs found
The Graph Curvature Calculator and the curvatures of cubic graphs
We classify all cubic graphs with either non-negative Ollivier-Ricci
curvature or non-negative Bakry-\'Emery curvature everywhere. We show in both
curvature notions that the non-negatively curved graphs are the prism graphs
and the M\"obius ladders. We also highlight an online tool for calculating the
curvature of graphs under several variants of these curvature notions that we
use in the classification. As a consequence of the classification result we
show, that non-negatively curved cubic expanders do not exist
Efficient and Robust Compressed Sensing Using Optimized Expander Graphs
Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any n-dimensional vector that is k-sparse can be fully recovered using O(klog n) measurements and only O(klog n) simple recovery iterations. In this paper, we improve upon this result by considering expander graphs with expansion coefficient beyond 3/4 and show that, with the same number of measurements, only O(k) recovery iterations are required, which is a significant improvement when n is large. In fact, full recovery can be accomplished by at most 2k very simple iterations. The number of iterations can be reduced arbitrarily close to k, and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time O(nlog(n/k))). We also show that by tolerating a small penal- ty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expander-graph-based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery. Finally, we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost k-sparse signal and then, using very simple optimization techniques, finds a k-sparse signal which is close to the best k-term approximation of the original signal
Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling
The goal of decentralized optimization over a network is to optimize a global
objective formed by a sum of local (possibly nonsmooth) convex functions using
only local computation and communication. It arises in various application
domains, including distributed tracking and localization, multi-agent
co-ordination, estimation in sensor networks, and large-scale optimization in
machine learning. We develop and analyze distributed algorithms based on dual
averaging of subgradients, and we provide sharp bounds on their convergence
rates as a function of the network size and topology. Our method of analysis
allows for a clear separation between the convergence of the optimization
algorithm itself and the effects of communication constraints arising from the
network structure. In particular, we show that the number of iterations
required by our algorithm scales inversely in the spectral gap of the network.
The sharpness of this prediction is confirmed both by theoretical lower bounds
and simulations for various networks. Our approach includes both the cases of
deterministic optimization and communication, as well as problems with
stochastic optimization and/or communication.Comment: 40 pages, 4 figure
Unconstraining Graph-Constrained Group Testing
In network tomography, one goal is to identify a small set of failed links in a network using as little information as possible. One way of setting up this problem is called graph-constrained group testing. Graph-constrained group testing is a variant of the classical combinatorial group testing problem, where the tests that one is allowed are additionally constrained by a graph. In this case, the graph is given by the underlying network topology.
The main contribution of this work is to show that for most graphs, the constraints imposed by the graph are no constraint at all. That is, the number of tests required to identify the failed links in graph-constrained group testing is near-optimal even for the corresponding group testing problem with no graph constraints. Our approach is based on a simple randomized construction of tests. To analyze our construction, we prove new results about the size of giant components in randomly sparsified graphs.
Finally, we provide empirical results which suggest that our connected-subgraph tests perform better not just in theory but also in practice, and in particular perform better on a real-world network topology
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