1,091 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
Efficient implementation of Radau collocation methods
In this paper we define an efficient implementation of Runge-Kutta methods of
Radau IIA type, which are commonly used when solving stiff ODE-IVPs problems.
The proposed implementation relies on an alternative low-rank formulation of
the methods, for which a splitting procedure is easily defined. The linear
convergence analysis of this splitting procedure exhibits excellent properties,
which are confirmed by its performance on a few numerical tests.Comment: 19 pages, 3 figures, 9 table
Quantum mereology: Factorizing Hilbert space into subsystems with quasiclassical dynamics
We study the question of how to decompose Hilbert space into a preferred tensor-product factorization without any preexisting structure other than a Hamiltonian operator, in particular the case of a bipartite decomposition into “system” and “environment.” Such a decomposition can be defined by looking for subsystems that exhibit quasiclassical behavior. The correct decomposition is one in which pointer states of the system are relatively robust against environmental monitoring (their entanglement with the environment does not continually and dramatically increase) and remain localized around approximately classical trajectories. We present an in-principle algorithm for finding such a decomposition by minimizing a combination of entanglement growth and internal spreading of the system. Both of these properties are related to locality in different ways. This formalism is relevant to questions in the foundations of quantum mechanics and the emergence of spacetime from quantum entanglement
Factorization and the Dressing Method for the Gel'fand-Dikii Hierarch
The isospectral flows of an order linear scalar differential
operator under the hypothesis that it possess a Baker-Akhiezer function
were originally investigated by Segal and Wilson from the point of view of
infinite dimensional Grassmanians, and the reduction of the KP hierarchy to the
Gel'fand-Dikii hierarchy. The associated first order systems and their formal
asymptotic solutions have a rich Lie algebraic structure which was investigated
by Drinfeld and Sokolov. We investigate the matrix Riemann-Hilbert
factorizations for these systems, and show that different factorizations lead
respectively to the potential, modified, and ordinary Gel'fand-Dikii flows. Lie
algebra decompositions (the Adler-Kostant-Symes method) are obtained for the
modified and potential flows. For the appropriate factorization for the
Gel'fand-Dikii flows is not a group factorization, as would be expected; yet a
modification of the dressing method still works.
A direct proof, based on a Fredholm determinant associated with the
factorization problem, is given that the potentials are meromorphic in and
in the time variables. Potentials with Baker-Akhiezer functions include the
multisoliton and rational solutions, as well as potentials in the scattering
class with compactly supported scattering data. The latter are dense in the
scattering class
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or
implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k))
floating-point operations (flops) in contrast to O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data
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