Seeded Graph Matching via Large Neighborhood Statistics

We study a well known noisy model of the graph isomorphism problem. In this model, the goal is to perfectly recover the vertex correspondence between two edge-correlated Erd\H{o}s-R\'{e}nyi random graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. For seeded problems, our result provides a significant improvement over previously known results. We show that it is possible to achieve the information-theoretic limit of graph sparsity in time polynomial in the number of vertices $n$. Moreover, we show the number of seeds needed for exact recovery in polynomial-time can be as low as $n^{3\epsilon}$ in the sparse graph regime (with the average degree smaller than $n^{\epsilon}$) and $\Omega(\log n)$ in the dense graph regime. Our results also shed light on the unseeded problem. In particular, we give sub-exponential time algorithms for sparse models and an $n^{O(\log n)}$ algorithm for dense models for some parameters, including some that are not covered by recent results of Barak et al

Efficient Compressive Sensing with Deterministic Guarantees Using Expander Graphs

Compressive sensing is an emerging technology which can recover a sparse signal vector of dimension n via a much smaller number of measurements than n. However, the existing compressive sensing methods may still suffer from relatively high recovery complexity, such as O(n^3), or can only work efficiently when the signal is super sparse, sometimes without deterministic performance guarantees. In this paper, we propose a compressive sensing scheme with deterministic performance guarantees using expander-graphs-based measurement matrices and show that the signal recovery can be achieved with complexity O(n) even if the number of nonzero elements k grows linearly with n. We also investigate compressive sensing for approximately sparse signals using this new method. Moreover, explicit constructions of the considered expander graphs exist. Simulation results are given to show the performance and complexity of the new method

Graph learning under sparsity priors

Graph signals offer a very generic and natural representation for data that lives on networks or irregular structures. The actual data structure is however often unknown a priori but can sometimes be estimated from the knowledge of the application domain. If this is not possible, the data structure has to be inferred from the mere signal observations. This is exactly the problem that we address in this paper, under the assumption that the graph signals can be represented as a sparse linear combination of a few atoms of a structured graph dictionary. The dictionary is constructed on polynomials of the graph Laplacian, which can sparsely represent a general class of graph signals composed of localized patterns on the graph. We formulate a graph learning problem, whose solution provides an ideal fit between the signal observations and the sparse graph signal model. As the problem is non-convex, we propose to solve it by alternating between a signal sparse coding and a graph update step. We provide experimental results that outline the good graph recovery performance of our method, which generally compares favourably to other recent network inference algorithms

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