6,476 research outputs found
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds
Descent methods for Nonnegative Matrix Factorization
In this paper, we present several descent methods that can be applied to
nonnegative matrix factorization and we analyze a recently developped fast
block coordinate method called Rank-one Residue Iteration (RRI). We also give a
comparison of these different methods and show that the new block coordinate
method has better properties in terms of approximation error and complexity. By
interpreting this method as a rank-one approximation of the residue matrix, we
prove that it \emph{converges} and also extend it to the nonnegative tensor
factorization and introduce some variants of the method by imposing some
additional controllable constraints such as: sparsity, discreteness and
smoothness.Comment: 47 pages. New convergence proof using damped version of RRI. To
appear in Numerical Linear Algebra in Signals, Systems and Control. Accepted.
Illustrating Matlab code is included in the source bundl
On the Simulatability Condition in Key Generation Over a Non-authenticated Public Channel
Simulatability condition is a fundamental concept in studying key generation
over a non-authenticated public channel, in which Eve is active and can
intercept, modify and falsify messages exchanged over the non-authenticated
public channel. Using this condition, Maurer and Wolf showed a remarkable "all
or nothing" result: if the simulatability condition does not hold, the key
capacity over the non-authenticated public channel will be the same as that of
the case with a passive Eve, while the key capacity over the non-authenticated
channel will be zero if the simulatability condition holds. However, two
questions remain open so far: 1) For a given joint probability mass function
(PMF), are there efficient algorithms (polynomial complexity algorithms) for
checking whether the simulatability condition holds or not?; and 2) If the
simulatability condition holds, are there efficient algorithms for finding the
corresponding attack strategy? In this paper, we answer these two open
questions affirmatively. In particular, for a given joint PMF, we construct a
linear programming (LP) problem and show that the simulatability condition
holds \textit{if and only if} the optimal value obtained from the constructed
LP is zero. Furthermore, we construct another LP and show that the minimizer of
the newly constructed LP is a valid attack strategy. Both LPs can be solved
with a polynomial complexity
Using Underapproximations for Sparse Nonnegative Matrix Factorization
Nonnegative Matrix Factorization consists in (approximately) factorizing a
nonnegative data matrix by the product of two low-rank nonnegative matrices. It
has been successfully applied as a data analysis technique in numerous domains,
e.g., text mining, image processing, microarray data analysis, collaborative
filtering, etc.
We introduce a novel approach to solve NMF problems, based on the use of an
underapproximation technique, and show its effectiveness to obtain sparse
solutions. This approach, based on Lagrangian relaxation, allows the resolution
of NMF problems in a recursive fashion. We also prove that the
underapproximation problem is NP-hard for any fixed factorization rank, using a
reduction of the maximum edge biclique problem in bipartite graphs.
We test two variants of our underapproximation approach on several standard
image datasets and show that they provide sparse part-based representations
with low reconstruction error. Our results are comparable and sometimes
superior to those obtained by two standard Sparse Nonnegative Matrix
Factorization techniques.Comment: Version 2 removed the section about convex reformulations, which was
not central to the development of our main results; added material to the
introduction; added a review of previous related work (section 2.3);
completely rewritten the last part (section 4) to provide extensive numerical
results supporting our claims. Accepted in J. of Pattern Recognitio
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