71,142 research outputs found
Alternative sampling for variational quantum Monte Carlo
Expectation values of physical quantities may accurately be obtained by the
evaluation of integrals within Many-Body Quantum mechanics, and these
multi-dimensional integrals may be estimated using Monte Carlo methods. In a
previous publication it has been shown that for the simplest, most commonly
applied strategy in continuum Quantum Monte Carlo, the random error in the
resulting estimates is not well controlled. At best the Central Limit theorem
is valid in its weakest form, and at worst it is invalid and replaced by an
alternative Generalised Central Limit theorem and non-Normal random error. In
both cases the random error is not controlled. Here we consider a new `residual
sampling strategy' that reintroduces the Central Limit Theorem in its strongest
form, and provides full control of the random error in estimates. Estimates of
the total energy and the variance of the local energy within Variational Monte
Carlo are considered in detail, and the approach presented may be generalised
to expectation values of other operators, and to other variants of the Quantum
Monte Carlo method.Comment: 14 pages, 9 figure
Conditional predictive inference post model selection
We give a finite-sample analysis of predictive inference procedures after
model selection in regression with random design. The analysis is focused on a
statistically challenging scenario where the number of potentially important
explanatory variables can be infinite, where no regularity conditions are
imposed on unknown parameters, where the number of explanatory variables in a
"good" model can be of the same order as sample size and where the number of
candidate models can be of larger order than sample size. The performance of
inference procedures is evaluated conditional on the training sample. Under
weak conditions on only the number of candidate models and on their complexity,
and uniformly over all data-generating processes under consideration, we show
that a certain prediction interval is approximately valid and short with high
probability in finite samples, in the sense that its actual coverage
probability is close to the nominal one and in the sense that its length is
close to the length of an infeasible interval that is constructed by actually
knowing the "best" candidate model. Similar results are shown to hold for
predictive inference procedures other than prediction intervals like, for
example, tests of whether a future response will lie above or below a given
threshold.Comment: Published in at http://dx.doi.org/10.1214/08-AOS660 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Statistical topological data analysis using persistence landscapes
We define a new topological summary for data that we call the persistence
landscape. Since this summary lies in a vector space, it is easy to combine
with tools from statistics and machine learning, in contrast to the standard
topological summaries. Viewed as a random variable with values in a Banach
space, this summary obeys a strong law of large numbers and a central limit
theorem. We show how a number of standard statistical tests can be used for
statistical inference using this summary. We also prove that this summary is
stable and that it can be used to provide lower bounds for the bottleneck and
Wasserstein distances.Comment: 26 pages, final version, to appear in Journal of Machine Learning
Research, includes two additional examples not in the journal version: random
geometric complexes and Erdos-Renyi random clique complexe
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