50,388 research outputs found
Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods
In this paper, we study matrix scaling and balancing, which are fundamental
problems in scientific computing, with a long line of work on them that dates
back to the 1960s. We provide algorithms for both these problems that, ignoring
logarithmic factors involving the dimension of the input matrix and the size of
its entries, both run in time where is the amount of error we are willing to
tolerate. Here, represents the ratio between the largest and the
smallest entries of the optimal scalings. This implies that our algorithms run
in nearly-linear time whenever is quasi-polynomial, which includes, in
particular, the case of strictly positive matrices. We complement our results
by providing a separate algorithm that uses an interior-point method and runs
in time .
In order to establish these results, we develop a new second-order
optimization framework that enables us to treat both problems in a unified and
principled manner. This framework identifies a certain generalization of linear
system solving that we can use to efficiently minimize a broad class of
functions, which we call second-order robust. We then show that in the context
of the specific functions capturing matrix scaling and balancing, we can
leverage and generalize the work on Laplacian system solving to make the
algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201
Renormalization group flows of Hamiltonians using tensor networks
A renormalization group flow of Hamiltonians for two-dimensional classical
partition functions is constructed using tensor networks. Similar to tensor
network renormalization ([G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405
(2015)], [S. Yang, Z.-C. Gu, and X.-G Wen, Phys. Rev. Lett. 118, 110504
(2017)]) we obtain approximate fixed point tensor networks at criticality. Our
formalism however preserves positivity of the tensors at every step and hence
yields an interpretation in terms of Hamiltonian flows. We emphasize that the
key difference between tensor network approaches and Kadanoff's spin blocking
method can be understood in terms of a change of local basis at every
decimation step, a property which is crucial to overcome the area law of mutual
information. We derive algebraic relations for fixed point tensors, calculate
critical exponents, and benchmark our method on the Ising model and the
six-vertex model.Comment: accepted version for Phys. Rev. Lett, main text: 5 pages, 3 figures,
appendices: 9 pages, 1 figur
Piecewise Parabolic Method on a Local Stencil for Magnetized Supersonic Turbulence Simulation
Stable, accurate, divergence-free simulation of magnetized supersonic
turbulence is a severe test of numerical MHD schemes and has been surprisingly
difficult to achieve due to the range of flow conditions present. Here we
present a new, higher order-accurate, low dissipation numerical method which
requires no additional dissipation or local "fixes" for stable execution. We
describe PPML, a local stencil variant of the popular PPM algorithm for solving
the equations of compressible ideal magnetohydrodynamics. The principal
difference between PPML and PPM is that cell interface states are evolved
rather that reconstructed at every timestep, resulting in a compact stencil.
Interface states are evolved using Riemann invariants containing all transverse
derivative information. The conservation laws are updated in an unsplit
fashion, making the scheme fully multidimensional. Divergence-free evolution of
the magnetic field is maintained using the higher order-accurate constrained
transport technique of Gardiner and Stone. The accuracy and stability of the
scheme is documented against a bank of standard test problems drawn from the
literature. The method is applied to numerical simulation of supersonic MHD
turbulence, which is important for many problems in astrophysics, including
star formation in dark molecular clouds. PPML accurately reproduces in
three-dimensions a transition to turbulence in highly compressible isothermal
gas in a molecular cloud model. The low dissipation and wide spectral bandwidth
of this method make it an ideal candidate for direct turbulence simulations.Comment: 28 pages, 18 figure
Half-integrality, LP-branching and FPT Algorithms
A recent trend in parameterized algorithms is the application of polytope
tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011;
Narayanaswamy et al., 2012). However, although interesting results have been
achieved, the methods require the underlying polytope to have very restrictive
properties (half-integrality and persistence), which are known only for few
problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node
Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we
view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a
relaxation of the search space from to such that
the new problem admits a polynomial-time exact solution. Using tools from CSP
(in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such
relaxations, we provide a much broader class of half-integral polytopes with
the required properties, unifying and extending previously known cases.
In addition to the insight into problems with half-integral relaxations, our
results yield a range of new and improved FPT algorithms, including an
-time algorithm for node-deletion Unique Label Cover with
label set and an -time algorithm for Group Feedback Vertex
Set, including the setting where the group is only given by oracle access. All
these significantly improve on previous results. The latter result also implies
the first single-exponential time FPT algorithm for Subset Feedback Vertex Set,
answering an open question of Cygan et al. (2012).
Additionally, we propose a network flow-based approach to solve some cases of
the relaxation problem. This gives the first linear-time FPT algorithm to
edge-deletion Unique Label Cover.Comment: Added results on linear-time FPT algorithms (not present in SODA
paper
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