105,785 research outputs found
On Constructing Spanners from Random Gaussian Projections
Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA\u2712) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area.
We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA\u2721), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work
Learning Graphs from Linear Measurements: Fundamental Trade-offs and Applications
We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We present a sparsity characterization for distributions of random graphs (that are allowed to contain high-degree nodes), based on which we study fundamental trade-offs between the number of measurements, the complexity of the graph class, and the probability of error. We first derive a necessary condition on the number of measurements. Then, by considering a three-stage recovery scheme, we give a sufficient condition for recovery. Furthermore, assuming the measurements are Gaussian IID, we prove upper and lower bounds on the (worst-case) sample complexity for both noisy and noiseless recovery. In the special cases of the uniform distribution on trees with n nodes and the Erdős-Rényi (n,p) class, the fundamental trade-offs are tight up to multiplicative factors with noiseless measurements. In addition, for practical applications, we design and implement a polynomial-time (in n ) algorithm based on the three-stage recovery scheme. Experiments show that the heuristic algorithm outperforms basis pursuit on star graphs. We apply the heuristic algorithm to learn admittance matrices in electric grids. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness and robustness of the proposed algorithm for parameter reconstruction
Information flow and cooperative control of vehicle formations
We consider the problem of cooperation among a collection of vehicles performing a shared task using intervehicle communication to coordinate their actions. Tools from algebraic graph theory prove useful in modeling the communication network and relating its topology to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the effect of the communication topology on formation stability. We also propose a method for decentralized information exchange between vehicles. This approach realizes a dynamical system that supplies each vehicle with a common reference to be used for cooperative motion. We prove a separation principle that decomposes formation stability into two components: Stability of this is achieved information flow for the given graph and stability of an individual vehicle for the given controller. The information flow can thus be rendered highly robust to changes in the graph, enabling tight formation control despite limitations in intervehicle communication capability
Complexity Analysis and Efficient Measurement Selection Primitives for High-Rate Graph SLAM
Sparsity has been widely recognized as crucial for efficient optimization in
graph-based SLAM. Because the sparsity and structure of the SLAM graph reflect
the set of incorporated measurements, many methods for sparsification have been
proposed in hopes of reducing computation. These methods often focus narrowly
on reducing edge count without regard for structure at a global level. Such
structurally-naive techniques can fail to produce significant computational
savings, even after aggressive pruning. In contrast, simple heuristics such as
measurement decimation and keyframing are known empirically to produce
significant computation reductions. To demonstrate why, we propose a
quantitative metric called elimination complexity (EC) that bridges the
existing analytic gap between graph structure and computation. EC quantifies
the complexity of the primary computational bottleneck: the factorization step
of a Gauss-Newton iteration. Using this metric, we show rigorously that
decimation and keyframing impose favorable global structures and therefore
achieve computation reductions on the order of and , respectively,
where is the pruning rate. We additionally present numerical results
showing EC provides a good approximation of computation in both batch and
incremental (iSAM2) optimization and demonstrate that pruning methods promoting
globally-efficient structure outperform those that do not.Comment: Pre-print accepted to ICRA 201
- …