108,225 research outputs found
Modified memoryless spectral-scaling Broyden family on Riemannian manifolds
This paper presents modified memoryless quasi-Newton methods based on the
spectral-scaling Broyden family on Riemannian manifolds. The method involves
adding one parameter to the search direction of the memoryless self-scaling
Broyden family on the manifold. Moreover, it uses a general map instead of
vector transport. This idea has already been proposed within a general
framework of Riemannian conjugate gradient methods where one can use vector
transport, scaled vector transport, or an inverse retraction. We show that the
search direction satisfies the sufficient descent condition under some
assumptions on the parameters. In addition, we show global convergence of the
proposed method under the Wolfe conditions. We numerically compare it with
existing methods, including Riemannian conjugate gradient methods and the
memoryless spectral-scaling Broyden family. The numerical results indicate that
the proposed method with the BFGS formula is suitable for solving an
off-diagonal cost function minimization problem on an oblique manifold.Comment: 20 pages, 8 figure
Optimization algorithms for the solution of the frictionless normal contact between rough surfaces
This paper revisits the fundamental equations for the solution of the
frictionless unilateral normal contact problem between a rough rigid surface
and a linear elastic half-plane using the boundary element method (BEM). After
recasting the resulting Linear Complementarity Problem (LCP) as a convex
quadratic program (QP) with nonnegative constraints, different optimization
algorithms are compared for its solution: (i) a Greedy method, based on
different solvers for the unconstrained linear system (Conjugate Gradient CG,
Gauss-Seidel, Cholesky factorization), (ii) a constrained CG algorithm, (iii)
the Alternating Direction Method of Multipliers (ADMM), and () the
Non-Negative Least Squares (NNLS) algorithm, possibly warm-started by
accelerated gradient projection steps or taking advantage of a loading history.
The latter method is two orders of magnitude faster than the Greedy CG method
and one order of magnitude faster than the constrained CG algorithm. Finally,
we propose another type of warm start based on a refined criterion for the
identification of the initial trial contact domain that can be used in
conjunction with all the previous optimization algorithms. This method, called
Cascade Multi-Resolution (CMR), takes advantage of physical considerations
regarding the scaling of the contact predictions by changing the surface
resolution. The method is very efficient and accurate when applied to real or
numerically generated rough surfaces, provided that their power spectral
density function is of power-law type, as in case of self-similar fractal
surfaces.Comment: 38 pages, 11 figure
Matrix product states and variational methods applied to critical quantum field theory
We study the second-order quantum phase-transition of massive real scalar
field theory with a quartic interaction ( theory) in (1+1) dimensions
on an infinite spatial lattice using matrix product states (MPS). We introduce
and apply a naive variational conjugate gradient method, based on the
time-dependent variational principle (TDVP) for imaginary time, to obtain
approximate ground states, using a related ansatz for excitations to calculate
the particle and soliton masses and to obtain the spectral density. We also
estimate the central charge using finite-entanglement scaling. Our value for
the critical parameter agrees well with recent Monte Carlo results, improving
on an earlier study which used the related DMRG method, verifying that these
techniques are well-suited to studying critical field systems. We also obtain
critical exponents that agree, as expected, with those of the transverse Ising
model. Additionally, we treat the special case of uniform product states (mean
field theory) separately, showing that they may be used to investigate
non-critical quantum field theories under certain conditions.Comment: 24 pages, 21 figures, with a minor improvement to the QFT sectio
Towards Scalable Spectral Clustering via Spectrum-Preserving Sparsification
Eigenvalue decomposition of Laplacian matrices for large nearest-neighbor (NN)graphs is the major computational bottleneck in spectral clustering (SC). To fundamentally address this computational challenge in SC, we propose a scalable spectral sparsification framework that enables to construct nearly-linear-sized ultra-sparse NN graphs with guaranteed preservation of key eigenvalues and eigenvectors of the original Laplacian. The proposed method is based on the latest theoretical results in spectral graph theory and thus can be applied to robustly handle general undirected graphs. By leveraging a nearly-linear time spectral graph topology sparsification phase and a subgraph scaling phase via stochastic gradient descent (SGD) iterations, our approach allows computing tree-like NN graphs that can serve as high-quality proxies of the original NN graphs, leading to highly-scalable and accurate SC of large data sets. Our extensive experimental results on a variety of public domain data sets show dramatically improved performance when compared with state-of-the-art SC methods
Multiclass Semi-Supervised Learning on Graphs using Ginzburg-Landau Functional Minimization
We present a graph-based variational algorithm for classification of
high-dimensional data, generalizing the binary diffuse interface model to the
case of multiple classes. Motivated by total variation techniques, the method
involves minimizing an energy functional made up of three terms. The first two
terms promote a stepwise continuous classification function with sharp
transitions between classes, while preserving symmetry among the class labels.
The third term is a data fidelity term, allowing us to incorporate prior
information into the model in a semi-supervised framework. The performance of
the algorithm on synthetic data, as well as on the COIL and MNIST benchmark
datasets, is competitive with state-of-the-art graph-based multiclass
segmentation methods.Comment: 16 pages, to appear in Springer's Lecture Notes in Computer Science
volume "Pattern Recognition Applications and Methods 2013", part of series on
Advances in Intelligent and Soft Computin
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