465 research outputs found
Randomly sparsified Richardson iteration is really fast
Recently, a class of algorithms combining classical fixed point iterations
with repeated random sparsification of approximate solution vectors has been
successfully applied to eigenproblems with matrices as large as . So far, a complete mathematical explanation for their success
has proven elusive. Additionally, the methods have not been extended to linear
system solves.
In this paper we propose a new scheme based on repeated random sparsification
that is capable of solving linear systems in extremely high dimensions. We
provide a complete mathematical analysis of this new algorithm. Our analysis
establishes a faster-than-Monte Carlo convergence rate and justifies use of the
scheme even when the solution vector itself is too large to store.Comment: 27 pages, 2 figure
Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo
Variational Monte Carlo (VMC) is an approach for computing ground-state
wavefunctions that has recently become more powerful due to the introduction of
neural network-based wavefunction parametrizations. However, efficiently
training neural wavefunctions to converge to an energy minimum remains a
difficult problem. In this work, we analyze optimization and sampling methods
used in VMC and introduce alterations to improve their performance. First,
based on theoretical convergence analysis in a noiseless setting, we motivate a
new optimizer that we call the Rayleigh-Gauss-Newton method, which can improve
upon gradient descent and natural gradient descent to achieve superlinear
convergence with little added computational cost. Second, in order to realize
this favorable comparison in the presence of stochastic noise, we analyze the
effect of sampling error on VMC parameter updates and experimentally
demonstrate that it can be reduced by the parallel tempering method. In
particular, we demonstrate that RGN can be made robust to energy spikes that
occur when new regions of configuration space become available to the sampler
over the course of optimization. Finally, putting theory into practice, we
apply our enhanced optimization and sampling methods to the transverse-field
Ising and XXZ models on large lattices, yielding ground-state energy estimates
with remarkably high accuracy after just 200-500 parameter updates.Comment: 12 pages, 7 figure
Randomized algorithms for low-rank matrix approximation: Design, analysis, and applications
This survey explores modern approaches for computing low-rank approximations
of high-dimensional matrices by means of the randomized SVD, randomized
subspace iteration, and randomized block Krylov iteration. The paper compares
the procedures via theoretical analyses and numerical studies to highlight how
the best choice of algorithm depends on spectral properties of the matrix and
the computational resources available.
Despite superior performance for many problems, randomized block Krylov
iteration has not been widely adopted in computational science. The paper
strengthens the case for this method in three ways. First, it presents new
pseudocode that can significantly reduce computational costs. Second, it
provides a new analysis that yields simple, precise, and informative error
bounds. Last, it showcases applications to challenging scientific problems,
including principal component analysis for genetic data and spectral clustering
for molecular dynamics data.Comment: 60 pages, 14 figure
Improved Fast Randomized Iteration Approach to Full Configuration Interaction
We present three modifications to our recently introduced fast randomized
iteration method for full configuration interaction (FCI-FRI) and investigate
their effects on the method's performance for Ne, HO, and N. The
initiator approximation, originally developed for full configuration
interaction quantum Monte Carlo, significantly reduces statistical error in
FCI-FRI when few samples are used in compression operations, enabling its
application to larger chemical systems. The semi-stochastic extension, which
involves exactly preserving a fixed subset of elements in each compression,
improves statistical efficiency in some cases but reduces it in others. We also
developed a new approach to sampling excitations that yields consistent
improvements in statistical efficiency and reductions in computational cost. We
discuss possible strategies based on our findings for improving the performance
of stochastic quantum chemistry methods more generally.Comment: 13 pages, 5 figure
Approximating matrix eigenvalues by subspace iteration with repeated random sparsification
Traditional numerical methods for calculating matrix eigenvalues are
prohibitively expensive for high-dimensional problems. Iterative random
sparsification methods allow for the estimation of a single dominant eigenvalue
at reduced cost by leveraging repeated random sampling and averaging. We
present a general approach to extending such methods for the estimation of
multiple eigenvalues and demonstrate its performance for several benchmark
problems in quantum chemistry.Comment: 31 pages, 7 figure
Understanding and eliminating spurious modes in variational Monte Carlo using collective variables
The use of neural network parametrizations to represent the ground state in
variational Monte Carlo (VMC) calculations has generated intense interest in
recent years. However, as we demonstrate in the context of the periodic
Heisenberg spin chain, this approach can produce unreliable wave function
approximations. One of the most obvious signs of failure is the occurrence of
random, persistent spikes in the energy estimate during training. These energy
spikes are caused by regions of configuration space that are over-represented
by the wave function density, which are called ``spurious modes'' in the
machine learning literature. After exploring these spurious modes in detail, we
demonstrate that a collective-variable-based penalization yields a
substantially more robust training procedure, preventing the formation of
spurious modes and improving the accuracy of energy estimates. Because the
penalization scheme is cheap to implement and is not specific to the particular
model studied here, it can be extended to other applications of VMC where a
reasonable choice of collective variable is available.Comment: 12 pages, 13 figure
Error bounds for dynamical spectral estimation
Dynamical spectral estimation is a well-established numerical approach for
estimating eigenvalues and eigenfunctions of the Markov transition operator
from trajectory data. Although the approach has been widely applied in
biomolecular simulations, its error properties remain poorly understood. Here
we analyze the error of a dynamical spectral estimation method called "the
variational approach to conformational dynamics" (VAC). We bound the
approximation error and estimation error for VAC estimates. Our analysis
establishes VAC's convergence properties and suggests new strategies for tuning
VAC to improve accuracy.Comment: 34 pages, 7 figure
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