Because of the significant increase in size and complexity of the networks, the distributed computation of eigenvalues and eigenvectors of graph matrices has become very challenging and yet it remains as important as before. In this paper we develop efficient distributed algorithms to detect, with higher resolution, closely situated eigenvalues and corresponding eigenvectors of symmetric graph matrices. We model the system of graph spectral computation as physical systems with Lagrangian and Hamiltonian dynamics. The spectrum of Laplacian matrix, in particular, is framed as a classical spring-mass system with Lagrangian dynamics. The spectrum of any general symmetric graph matrix turns out to have a simple connection with quantum systems and it can be thus formulated as a solution to a Schr "odinger-type differential equation. Taking into account the higher resolution requirement in the spectrum computation and the related stability issues in the numerical solution of the underlying differential equation, we propose the application of symplectic integrators to the calculation of eigenspectrum. The effectiveness of the proposed techniques is demonstrated with numerical simulations on real-world networks of different sizes and complexities.
In comparison to @cite_7 we do not deform the system and we use new symplectic numerical integrators @cite_1 @cite_12 . For the problem of distributed spectral decomposition in networks, one of the first and prominent works appeared in @cite_5 . But their algorithm requires distributed orthonormalization at each step and they solve this difficult operation via random walks. But if the graph is not well-connected (low conductance), this task will take a very long time to converge. Our distributed implementation based on fluid diffusion in the network does not require such orthonormalization.
@cite_5 @cite_1 @cite_12 @cite_7
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The authors introduce a concept of network decomposition, a partitioning of an arbitrary graph into small-diameter connected components, such that the graph created by contracting each component into a single node has low chromatic number. They present an efficient distributed algorithm for constructing such a decomposition and demonstrate its use for design of efficient distributed algorithms. The method yields new deterministic distributed algorithms for finding a maximal independent set in an arbitrary graph and for ( Delta +1)-coloring of graphs with maximum degree Delta . These algorithms run in O(n sup epsilon ) time for epsilon =O((log log n log n) sup 1 2 ), whereas the best previously known deterministic algorithms required Omega (n) time. The techniques can also be used to remove randomness from the previously known most distributed breadth-first search algorithm. > We give an overview of recent developments in the problem of reconstructing a band-limited signal from nonuniform sampling from a numerical analysis view point. It is shown that the appropriate design of the finite-dimensional model plays a key role in the numerical solution of the nonuniform sampling problem. In the one approach (often proposed in the literature) the finite-dimensional model leads to an ill-posed problem even in very simple situations. The other approach that we consider leads to a well-posed problem that preserves important structural properties of the original infinite-dimensional problem and gives rise to efficient numerical algorithms. Furthermore, a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy nonuniformly spaced samples. We also discuss the design of efficient regularization methods for ill-conditioned reconstruction problems. Numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods. This paper describes a novel approach of packing sparse convolutional neural networks for their efficient systolic array implementations. By combining subsets of columns in the original filter matrix associated with a convolutional layer, we increase the utilization efficiency of the systolic array substantially (e.g., 4x) due to the increased density of nonzeros in the resulting packed filter matrix. In combining columns, for each row, all filter weights but one with the largest magnitude are pruned. We retrain the remaining weights to preserve high accuracy. We demonstrate that in mitigating data privacy concerns the retraining can be accomplished with only fractions of the original dataset (e.g., 10 for CIFAR-10). We study the effectiveness of this joint optimization for both high utilization and classification accuracy with ASIC and FPGA designs based on efficient bit-serial implementations of multiplier-accumulators. We present analysis and empirical evidence on the superior performance of our column combining approach against prior arts under metrics such as energy efficiency (3x) and inference latency (12x). We study the problem of computing a minimum cut in a simple, undirected graph and give a deterministic O(m log2 n log log2 n) time algorithm. This improves both on the best previously known deterministic running time of O(m log12 n) (Kawarabayashi and Thorup [12]) and the best previously known randomized running time of O(m log3 n) (Karger [11]) for this problem, though Karger's algorithm can be further applied to weighted graphs. Our approach is using the Kawarabayashi and Thorup graph compression technique, which repeatedly finds low-conductance cuts. To find these cuts they use a diffusion-based local algorithm. We use instead a flow-based local algorithm and suitably adjust their framework to work with our flow-based subroutine. Both flow and diffusion based methods have a long history of being applied to finding low conductance cuts. Diffusion algorithms have several variants that are naturally local while it is more complicated to make flow methods local. Some prior work has proven nice properties for local flow based algorithms with respect to improving or cleaning up low conductance cuts. Our flow subroutine, however, is the first that is both local and produces low conductance cuts. Thus, it may be of independent interest.