138,342 research outputs found
Statistical Optimality of Divide and Conquer Kernel-based Functional Linear Regression
Previous analysis of regularized functional linear regression in a
reproducing kernel Hilbert space (RKHS) typically requires the target function
to be contained in this kernel space. This paper studies the convergence
performance of divide-and-conquer estimators in the scenario that the target
function does not necessarily reside in the underlying RKHS. As a
decomposition-based scalable approach, the divide-and-conquer estimators of
functional linear regression can substantially reduce the algorithmic
complexities in time and memory. We develop an integral operator approach to
establish sharp finite sample upper bounds for prediction with
divide-and-conquer estimators under various regularity conditions of
explanatory variables and target function. We also prove the asymptotic
optimality of the derived rates by building the mini-max lower bounds. Finally,
we consider the convergence of noiseless estimators and show that the rates can
be arbitrarily fast under mild conditions
Non-asymptotic Optimal Prediction Error for RKHS-based Partially Functional Linear Models
Under the framework of reproducing kernel Hilbert space (RKHS), we consider
the penalized least-squares of the partially functional linear models (PFLM),
whose predictor contains both functional and traditional multivariate part, and
the multivariate part allows a divergent number of parameters. From the
non-asymptotic point of view, we focus on the rate-optimal upper and lower
bounds of the prediction error. An exact upper bound for the excess prediction
risk is shown in a non-asymptotic form under a more general assumption known as
the effective dimension to the model, by which we also show the prediction
consistency when the number of multivariate covariates slightly increases
with the sample size . Our new finding implies a trade-off between the
number of non-functional predictors and the effective dimension of the kernel
principal components to ensure the prediction consistency in the
increasing-dimensional setting. The analysis in our proof hinges on the
spectral condition of the sandwich operator of the covariance operator and the
reproducing kernel, and on the concentration inequalities for the random
elements in Hilbert space. Finally, we derive the non-asymptotic minimax lower
bound under the regularity assumption of Kullback-Leibler divergence of the
models.Comment: 24 page
Consistent Searches for SMEFT Effects in Non-Resonant Dilepton Events
Employing the framework of the Standard Model Effective Field Theory, we
perform a detailed reinterpretation of measurements of the Weinberg angle in
dilepton production as a search for new-physics effects. We truncate our signal
prediction at order , where denotes the new-physics mass
scale, and introduce a theory error to account for unknown contributions of
order . Two linear combinations of four-fermion operators with
distinct angular behavior contribute to dilepton production with growing impact
at high energies. We define suitable angular observables and derive bounds on
those two linear combinations using data from the Tevatron and the LHC. We find
that the current data is able to constrain interesting regions of parameter
space, with important contributions at lower cutoff scales from the Tevatron,
and that the future LHC data will eventually be able to simultaneously
constrain both independent linear combinations which contribute to dilepton
production.Comment: 11 pages, 4 figures v2: updated to match version accepted for
publicatio
The ROMES method for statistical modeling of reduced-order-model error
This work presents a technique for statistically modeling errors introduced
by reduced-order models. The method employs Gaussian-process regression to
construct a mapping from a small number of computationally inexpensive `error
indicators' to a distribution over the true error. The variance of this
distribution can be interpreted as the (epistemic) uncertainty introduced by
the reduced-order model. To model normed errors, the method employs existing
rigorous error bounds and residual norms as indicators; numerical experiments
show that the method leads to a near-optimal expected effectivity in contrast
to typical error bounds. To model errors in general outputs, the method uses
dual-weighted residuals---which are amenable to uncertainty control---as
indicators. Experiments illustrate that correcting the reduced-order-model
output with this surrogate can improve prediction accuracy by an order of
magnitude; this contrasts with existing `multifidelity correction' approaches,
which often fail for reduced-order models and suffer from the curse of
dimensionality. The proposed error surrogates also lead to a notion of
`probabilistic rigor', i.e., the surrogate bounds the error with specified
probability
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