19,984 research outputs found
Profile minimization on products of graphs
AbstractThe profile minimization problem arose from the study of sparse matrix technique. In terms of graphs, the problem is to determine the profile of a graph G which is defined asP(G)=minf∑v∈V(G)maxx∈N[v](f(v)-f(x)),where f runs over all bijections from V(G) to {1,2,…,|V(G)|} and N[v]={v}∪{x∈V(G):xv∈E(G)}. The main result of this paper is to determine the profiles of Km×Kn, Ks,t×Kn and Pm×Kn
Matrix Completion on Graphs
The problem of finding the missing values of a matrix given a few of its
entries, called matrix completion, has gathered a lot of attention in the
recent years. Although the problem under the standard low rank assumption is
NP-hard, Cand\`es and Recht showed that it can be exactly relaxed if the number
of observed entries is sufficiently large. In this work, we introduce a novel
matrix completion model that makes use of proximity information about rows and
columns by assuming they form communities. This assumption makes sense in
several real-world problems like in recommender systems, where there are
communities of people sharing preferences, while products form clusters that
receive similar ratings. Our main goal is thus to find a low-rank solution that
is structured by the proximities of rows and columns encoded by graphs. We
borrow ideas from manifold learning to constrain our solution to be smooth on
these graphs, in order to implicitly force row and column proximities. Our
matrix recovery model is formulated as a convex non-smooth optimization
problem, for which a well-posed iterative scheme is provided. We study and
evaluate the proposed matrix completion on synthetic and real data, showing
that the proposed structured low-rank recovery model outperforms the standard
matrix completion model in many situations.Comment: Version of NIPS 2014 workshop "Out of the Box: Robustness in High
Dimension
Adding Logical Operators to Tree Pattern Queries on Graph-Structured Data
As data are increasingly modeled as graphs for expressing complex
relationships, the tree pattern query on graph-structured data becomes an
important type of queries in real-world applications. Most practical query
languages, such as XQuery and SPARQL, support logical expressions using
logical-AND/OR/NOT operators to define structural constraints of tree patterns.
In this paper, (1) we propose generalized tree pattern queries (GTPQs) over
graph-structured data, which fully support propositional logic of structural
constraints. (2) We make a thorough study of fundamental problems including
satisfiability, containment and minimization, and analyze the computational
complexity and the decision procedures of these problems. (3) We propose a
compact graph representation of intermediate results and a pruning approach to
reduce the size of intermediate results and the number of join operations --
two factors that often impair the efficiency of traditional algorithms for
evaluating tree pattern queries. (4) We present an efficient algorithm for
evaluating GTPQs using 3-hop as the underlying reachability index. (5)
Experiments on both real-life and synthetic data sets demonstrate the
effectiveness and efficiency of our algorithm, from several times to orders of
magnitude faster than state-of-the-art algorithms in terms of evaluation time,
even for traditional tree pattern queries with only conjunctive operations.Comment: 16 page
Convex recovery from interferometric measurements
This note formulates a deterministic recovery result for vectors from
quadratic measurements of the form for some
left-invertible . Recovery is exact, or stable in the noisy case, when the
couples are chosen as edges of a well-connected graph. One possible way
of obtaining the solution is as a feasible point of a simple semidefinite
program. Furthermore, we show how the proportionality constant in the error
estimate depends on the spectral gap of a data-weighted graph Laplacian. Such
quadratic measurements have found applications in phase retrieval, angular
synchronization, and more recently interferometric waveform inversion
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