10,168 research outputs found
See the Tree Through the Lines: The Shazoo Algorithm -- Full Version --
Predicting the nodes of a given graph is a fascinating theoretical problem
with applications in several domains. Since graph sparsification via spanning
trees retains enough information while making the task much easier, trees are
an important special case of this problem. Although it is known how to predict
the nodes of an unweighted tree in a nearly optimal way, in the weighted case a
fully satisfactory algorithm is not available yet. We fill this hole and
introduce an efficient node predictor, Shazoo, which is nearly optimal on any
weighted tree. Moreover, we show that Shazoo can be viewed as a common
nontrivial generalization of both previous approaches for unweighted trees and
weighted lines. Experiments on real-world datasets confirm that Shazoo performs
well in that it fully exploits the structure of the input tree, and gets very
close to (and sometimes better than) less scalable energy minimization methods
Online Prediction of Switching Graph Labelings with Cluster Specialists
We address the problem of predicting the labeling of a graph in an online
setting when the labeling is changing over time. We present an algorithm based
on a specialist approach; we develop the machinery of cluster specialists which
probabilistically exploits the cluster structure in the graph. Our algorithm
has two variants, one of which surprisingly only requires
time on any trial on an -vertex graph, an exponential speed up over
existing methods. We prove switching mistake-bound guarantees for both variants
of our algorithm. Furthermore these mistake bounds smoothly vary with the
magnitude of the change between successive labelings. We perform experiments on
Chicago Divvy Bicycle Sharing data and show that our algorithms significantly
outperform an existing algorithm (a kernelized Perceptron) as well as several
natural benchmarks.Comment: 20 pages (including appendix
A Scalable Multiclass Algorithm for Node Classification
We introduce a scalable algorithm, MUCCA, for multiclass node classification
in weighted graphs. Unlike previously proposed methods for the same task, MUCCA
works in time linear in the number of nodes. Our approach is based on a
game-theoretic formulation of the problem in which the test labels are
expressed as a Nash Equilibrium of a certain game. However, in order to achieve
scalability, we find the equilibrium on a spanning tree of the original graph.
Experiments on real-world data reveal that MUCCA is much faster than its
competitors while achieving a similar predictive performance
Active Learning on Trees and Graphs
We investigate the problem of active learning on a given tree whose nodes are
assigned binary labels in an adversarial way. Inspired by recent results by
Guillory and Bilmes, we characterize (up to constant factors) the optimal
placement of queries so to minimize the mistakes made on the non-queried nodes.
Our query selection algorithm is extremely efficient, and the optimal number of
mistakes on the non-queried nodes is achieved by a simple and efficient mincut
classifier. Through a simple modification of the query selection algorithm we
also show optimality (up to constant factors) with respect to the trade-off
between number of queries and number of mistakes on non-queried nodes. By using
spanning trees, our algorithms can be efficiently applied to general graphs,
although the problem of finding optimal and efficient active learning
algorithms for general graphs remains open. Towards this end, we provide a
lower bound on the number of mistakes made on arbitrary graphs by any active
learning algorithm using a number of queries which is up to a constant fraction
of the graph size
Learning Structured Outputs from Partial Labels using Forest Ensemble
Learning structured outputs with general structures is computationally
challenging, except for tree-structured models. Thus we propose an efficient
boosting-based algorithm AdaBoost.MRF for this task. The idea is based on the
realization that a graph is a superimposition of trees. Different from most
existing work, our algorithm can handle partial labelling, and thus is
particularly attractive in practice where reliable labels are often sparsely
observed. In addition, our method works exclusively on trees and thus is
guaranteed to converge. We apply the AdaBoost.MRF algorithm to an indoor video
surveillance scenario, where activities are modelled at multiple levels.Comment: Conference version appeared in Truyen et al, AdaBoost.MRF: Boosted
Markov random forests and application to multilevel activity recognition.
CVPR'0
Approximate -penalized estimation of piecewise-constant signals on graphs
We study recovery of piecewise-constant signals on graphs by the estimator
minimizing an -edge-penalized objective. Although exact minimization of
this objective may be computationally intractable, we show that the same
statistical risk guarantees are achieved by the -expansion algorithm
which computes an approximate minimizer in polynomial time. We establish that
for graphs with small average vertex degree, these guarantees are minimax
rate-optimal over classes of edge-sparse signals. For spatially inhomogeneous
graphs, we propose minimization of an edge-weighted objective where each edge
is weighted by its effective resistance or another measure of its contribution
to the graph's connectivity. We establish minimax optimality of the resulting
estimators over corresponding edge-weighted sparsity classes. We show
theoretically that these risk guarantees are not always achieved by the
estimator minimizing the /total-variation relaxation, and empirically that
the -based estimates are more accurate in high signal-to-noise settings.Comment: v2: Title change, renumbering of sections and theorem
Graphical Models as Block-Tree Graphs
We introduce block-tree graphs as a framework for deriving efficient
algorithms on graphical models. We define block-tree graphs as a
tree-structured graph where each node is a cluster of nodes such that the
clusters in the graph are disjoint. This differs from junction-trees, where two
clusters connected by an edge always have at least one common node. When
compared to junction-trees, we show that constructing block-tree graphs is
faster, and finding optimal block-tree graphs has a much smaller search space.
Applying our block-tree graph framework to graphical models, we show that, for
some graphs, e.g., grid graphs, using block-tree graphs for inference is
computationally more efficient than using junction-trees. For graphical models
with boundary conditions, the block-tree graph framework transforms the
boundary valued problem into an initial value problem. For Gaussian graphical
models, the block-tree graph framework leads to a linear state-space
representation. Since exact inference in graphical models can be
computationally intractable, we propose to use spanning block-trees to derive
approximate inference algorithms. Experimental results show the improved
performance in using spanning block-trees versus using spanning trees for
approximate estimation over Gaussian graphical models.Comment: 29 pages. Correction to version
Deep learning long-range information in undirected graphs with wave networks
Graph algorithms are key tools in many fields of science and technology. Some
of these algorithms depend on propagating information between distant nodes in
a graph. Recently, there have been a number of deep learning architectures
proposed to learn on undirected graphs. However, most of these architectures
aggregate information in the local neighborhood of a node, and therefore they
may not be capable of efficiently propagating long-range information. To solve
this problem we examine a recently proposed architecture, wave, which
propagates information back and forth across an undirected graph in waves of
nonlinear computation. We compare wave to graph convolution, an architecture
based on local aggregation, and find that wave learns three different
graph-based tasks with greater efficiency and accuracy. These three tasks
include (1) labeling a path connecting two nodes in a graph, (2) solving a maze
presented as an image, and (3) computing voltages in a circuit. These tasks
range from trivial to very difficult, but wave can extrapolate from small
training examples to much larger testing examples. These results show that wave
may be able to efficiently solve a wide range of problems that require
long-range information propagation across undirected graphs. An implementation
of the wave network, and example code for the maze problem are included in the
tflon deep learning toolkit (https://bitbucket.org/mkmatlock/tflon)
Asymptotic Enumeration of Spanning Trees
We give new general formulas for the asymptotics of the number of spanning
trees of a large graph. A special case answers a question of McKay (1983) for
regular graphs. The general answer involves a quantity for infinite graphs that
we call "tree entropy", which we show is a logarithm of a normalized
determinant of the graph Laplacian for infinite graphs. Tree entropy is also
expressed using random walks. We relate tree entropy to the metric entropy of
the uniform spanning forest process on quasi-transitive amenable graphs,
extending a result of Burton and Pemantle (1993).Comment: 38 page
Active spanning trees and Schramm-Loewner evolution
We consider the Peano curve separating a spanning tree from its dual spanning
tree on an embedded planar graph, where the tree and dual tree are weighted by
to the number of active edges, and "active" is in the sense of the Tutte
polynomial. When the graph is a portion of the square grid approximating a
simply connected domain, it is known ( and ) or believed
() that the Peano curve converges to a space-filling SLE
loop, where , corresponding to . We
argue that the same should hold for , which corresponds to
.Comment: 6 pages, 7 figure
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