810 research outputs found
Efficient Numerical Methods to Solve Sparse Linear Equations with Application to PageRank
In this paper, we propose three methods to solve the PageRank problem for the
transition matrices with both row and column sparsity. Our methods reduce the
PageRank problem to the convex optimization problem over the simplex. The first
algorithm is based on the gradient descent in L1 norm instead of the Euclidean
one. The second algorithm extends the Frank-Wolfe to support sparse gradient
updates. The third algorithm stands for the mirror descent algorithm with a
randomized projection. We proof converges rates for these methods for sparse
problems as well as numerical experiments support their effectiveness.Comment: 26 page
Gossip Algorithms for Distributed Signal Processing
Gossip algorithms are attractive for in-network processing in sensor networks
because they do not require any specialized routing, there is no bottleneck or
single point of failure, and they are robust to unreliable wireless network
conditions. Recently, there has been a surge of activity in the computer
science, control, signal processing, and information theory communities,
developing faster and more robust gossip algorithms and deriving theoretical
performance guarantees. This article presents an overview of recent work in the
area. We describe convergence rate results, which are related to the number of
transmitted messages and thus the amount of energy consumed in the network for
gossiping. We discuss issues related to gossiping over wireless links,
including the effects of quantization and noise, and we illustrate the use of
gossip algorithms for canonical signal processing tasks including distributed
estimation, source localization, and compression.Comment: Submitted to Proceedings of the IEEE, 29 page
Similarity-Aware Spectral Sparsification by Edge Filtering
In recent years, spectral graph sparsification techniques that can compute
ultra-sparse graph proxies have been extensively studied for accelerating
various numerical and graph-related applications. Prior nearly-linear-time
spectral sparsification methods first extract low-stretch spanning tree from
the original graph to form the backbone of the sparsifier, and then recover
small portions of spectrally-critical off-tree edges to the spanning tree to
significantly improve the approximation quality. However, it is not clear how
many off-tree edges should be recovered for achieving a desired spectral
similarity level within the sparsifier. Motivated by recent graph signal
processing techniques, this paper proposes a similarity-aware spectral graph
sparsification framework that leverages efficient spectral off-tree edge
embedding and filtering schemes to construct spectral sparsifiers with
guaranteed spectral similarity (relative condition number) level. An iterative
graph densification scheme is introduced to facilitate efficient and effective
filtering of off-tree edges for highly ill-conditioned problems. The proposed
method has been validated using various kinds of graphs obtained from public
domain sparse matrix collections relevant to VLSI CAD, finite element analysis,
as well as social and data networks frequently studied in many machine learning
and data mining applications
Activity recognition from videos with parallel hypergraph matching on GPUs
In this paper, we propose a method for activity recognition from videos based
on sparse local features and hypergraph matching. We benefit from special
properties of the temporal domain in the data to derive a sequential and fast
graph matching algorithm for GPUs.
Traditionally, graphs and hypergraphs are frequently used to recognize
complex and often non-rigid patterns in computer vision, either through graph
matching or point-set matching with graphs. Most formulations resort to the
minimization of a difficult discrete energy function mixing geometric or
structural terms with data attached terms involving appearance features.
Traditional methods solve this minimization problem approximately, for instance
with spectral techniques.
In this work, instead of solving the problem approximatively, the exact
solution for the optimal assignment is calculated in parallel on GPUs. The
graphical structure is simplified and regularized, which allows to derive an
efficient recursive minimization algorithm. The algorithm distributes
subproblems over the calculation units of a GPU, which solves them in parallel,
allowing the system to run faster than real-time on medium-end GPUs
Spectral Complexity of Directed Graphs and Application to Structural Decomposition
We introduce a new measure of complexity (called spectral complexity) for
directed graphs. We start with splitting of the directed graph into its
recurrent and non-recurrent parts. We define the spectral complexity metric in
terms of the spectrum of the recurrence matrix (associated with the reccurent
part of the graph) and the Wasserstein distance. We show that the total
complexity of the graph can then be defined in terms of the spectral
complexity, complexities of individual components and edge weights. The
essential property of the spectral complexity metric is that it accounts for
directed cycles in the graph. In engineered and software systems, such cycles
give rise to sub-system interdependencies and increase risk for unintended
consequences through positive feedback loops, instabilities, and infinite
execution loops in software. In addition, we present a structural decomposition
technique that identifies such cycles using a spectral technique. We show that
this decomposition complements the well-known spectral decomposition analysis
based on the Fiedler vector. We provide several examples of computation of
spectral and total complexities, including the demonstration that the
complexity increases monotonically with the average degree of a random graph.
We also provide an example of spectral complexity computation for the
architecture of a realistic fixed wing aircraft system.Comment: We added new theoretical results in Section 2 and introduced a new
section 2.2 devoted to intuitive and physical explanations of the concepts
from the pape
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