236 research outputs found
Don't Fall for Tuning Parameters: Tuning-Free Variable Selection in High Dimensions With the TREX
Lasso is a seminal contribution to high-dimensional statistics, but it hinges
on a tuning parameter that is difficult to calibrate in practice. A partial
remedy for this problem is Square-Root Lasso, because it inherently calibrates
to the noise variance. However, Square-Root Lasso still requires the
calibration of a tuning parameter to all other aspects of the model. In this
study, we introduce TREX, an alternative to Lasso with an inherent calibration
to all aspects of the model. This adaptation to the entire model renders TREX
an estimator that does not require any calibration of tuning parameters. We
show that TREX can outperform cross-validated Lasso in terms of variable
selection and computational efficiency. We also introduce a bootstrapped
version of TREX that can further improve variable selection. We illustrate the
promising performance of TREX both on synthetic data and on a recent
high-dimensional biological data set that considers riboflavin production in B.
subtilis
Non-convex Global Minimization and False Discovery Rate Control for the TREX
The TREX is a recently introduced method for performing sparse
high-dimensional regression. Despite its statistical promise as an alternative
to the lasso, square-root lasso, and scaled lasso, the TREX is computationally
challenging in that it requires solving a non-convex optimization problem. This
paper shows a remarkable result: despite the non-convexity of the TREX problem,
there exists a polynomial-time algorithm that is guaranteed to find the global
minimum. This result adds the TREX to a very short list of non-convex
optimization problems that can be globally optimized (principal components
analysis being a famous example). After deriving and developing this new
approach, we demonstrate that (i) the ability of the preexisting TREX heuristic
to reach the global minimum is strongly dependent on the difficulty of the
underlying statistical problem, (ii) the new polynomial-time algorithm for TREX
permits a novel variable ranking and selection scheme, (iii) this scheme can be
incorporated into a rule that controls the false discovery rate (FDR) of
included features in the model. To achieve this last aim, we provide an
extension of the results of Barber & Candes (2015) to establish that the
knockoff filter framework can be applied to the TREX. This investigation thus
provides both a rare case study of a heuristic for non-convex optimization and
a novel way of exploiting non-convexity for statistical inference
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Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations
This paper improves guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the errors of divergence-free primal and H(div)-conforming dual mixed methods (for the velocity gradient) with an equilibration constraint that needs special care when discretised. To relax the constraints on the primal and dual method, a more general result is derived that enables the use of a recently developed mass conserving mixed stress discretisation to design equilibrated fluxes that yield pressure-independent guaranteed upper bounds for any pressure-robust (but not necessarily divergence-free) primal discretisation. Moreover, a provably efficient local design of the equilibrated fluxes is presented that reduces the numerical costs of the error estimator. All theoretical findings are verified by numerical examples which also show that the efficiency indices of our novel guaranteed upper bounds for the velocity error are close to 1
Gradient-robust hybrid DG discretizations for the compressible Stokes equations
This paper studies two hybrid discontinuous Galerkin (HDG) discretizations
for the velocity-density formulation of the compressible Stokes equations with
respect to several desired structural properties, namely provable convergence,
the preservation of non-negativity and mass constraints for the density, and
gradient-robustness. The later property dramatically enhances the accuracy in
well-balanced situations, such as the hydrostatic balance where the pressure
gradient balances the gravity force. One of the studied schemes employs an
H(div)-conforming velocity ansatz space which ensures all mentioned properties,
while a fully discontinuous method is shown to satisfy all properties but the
gradient-robustness. Also higher-order schemes for both variants are presented
and compared in three numerical benchmark problems. The final example shows the
importance also for non-hydrostatic well-balanced states for the compressible
Navier-Stokes equations
Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations
This paper improves guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the errors of divergence-free primal and H(div)-conforming dual mixed methods (for the velocity gradient) with an equilibration constraint that needs special care when discretised. To relax the constraints on the primal and dual method, a more general result is derived that enables the use of a recently developed mass conserving mixed stress discretisation to design equilibrated fluxes that yield pressure-independent guaranteed upper bounds for any pressure-robust (but not necessarily divergence-free) primal discretisation. Moreover, a provably efficient local design of the equilibrated fluxes is presented that reduces the numerical costs of the error estimator. All theoretical findings are verified by numerical examples which also show that the efficiency indices of our novel guaranteed upper bounds for the velocity error are close to 1
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Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods
Recent works showed that pressure-robust modifications of mixed finite
element methods for the Stokes equations outperform their standard versions
in many cases. This is achieved by divergence-free reconstruction operators
and results in pressure-independent velocity error estimates which are robust
with respect to small viscosities. In this paper we develop a posteriori
error control which reflects this robustness. The main difficulty lies in the
volume contribution of the standard residual-based approach that includes the
L2-norm of the right-hand side. However, the velocity is only steered by the
divergence-free part of this source term. An efficient error estimator must
approximate this divergence-free part in a proper manner, otherwise it can be
dominated by the pressure error. To overcome this difficulty a novel approach
is suggested that uses arguments from the stream function and vorticity
formulation of the NavierStokes equations. The novel error estimators only
take the curl of the righthand side into account and so lead to provably
reliable, efficient and pressure-independent upper bounds in case of a
pressure-robust method in particular in pressure-dominant situations. This is
also confirmed by some numerical examples with the novel pressure-robust
modifications of the TaylorHood and mini finite element methods
Long-term gait measurements in daily life: Results from the Berlin Aging Study II (BASE-II)
BACKGROUND:
Walking ability is an important prerequisite for activity, social participation and independent living. While in most healthy adults, this ability can be assumed as given, limitations in walking ability occur with increasing age. Furthermore, slow walking speed is linked to several chronic conditions and overall morbidity. Measurements of gait parameters can be used as a proxy to detect functional decline and onset of chronic conditions. Up to now, gait characteristics used for this purpose are measured in standardized laboratory settings. There is some evidence, however, that long-term measurements of gait parameters in the living environment have some advantages over short-term laboratory measurements.
METHODS:
We evaluated cross-sectional data from an accelerometric sensor worn in a subgroup of 554 participants of the Berlin Aging Study II (BASE-II). Data from the two BASE-II age groups (age between 22-36 years and 60-79 years) were used for the current analysis of accelerometric data for a minimum of two days and a maximum of ten days were available. Real world walking speed, number of steps, maximum coherent distance and total distance were derived as average data per day. Linear regression analyses were performed on the different gait parameters in order to identify significant determinants. Additionally, Mann-Whitney-U-tests were performed to detect sex-specific differences.
RESULTS:
Age showed to be significantly associated with real world walking speed and with the total distance covered per day, while BMI contributed negatively to the number of walking steps, maximum coherent distance and total distance walked. Additionally, sex was associated with walking steps. However, R2-values for all models were low. Overall, women had significantly more walking steps and a larger coherent distance per day when compared to men. When separated by age group, this difference was significant only in the older participants. Additionally, walking speed was significantly higher in women compared to men in the subgroup of older people.
CONCLUSIONS:
Age- and sex-specific differences have to be considered when objective gait parameters are measured, e.g. in the context of clinical risk assessment. For this purpose normative data, differentiating for age and sex would have to be established to allow reliable classification of long-term measurements of gait
Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements
Classical inf-sup stable mixed finite elements for the incompressible (Navier--)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. How-ever, a modification only in the right hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements with discontinuous discrete pressures. In this contribution, this idea is extended to low and high order Taylor--Hood and mini elements, which have continuous discrete pressures. For the modification of the right hand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. The reconstruction is based on local H(div)-conforming flux equilibration on vertex patches, and fulfills certain orthogonality properties to provide consistency and optimal a-priori error estimates. Numerical examples for the incompressible Stokes and Navier--Stokes equations confirm that the new pressure-robust Taylor--Hood and mini elements converge with optimal order and outperform signi--cantly the classical versions of those elements when the continuous pressure is comparably large
Automatic identification of chemical moieties
In recent years, the prediction of quantum mechanical observables with machine learning methods has become increasingly popular. Message-passing neural networks (MPNNs) solve this task by constructing atomic representations, from which the properties of interest are predicted. Here, we introduce a method to automatically identify chemical moieties (molecular building blocks) from such representations, enabling a variety of applications beyond property prediction, which otherwise rely on expert knowledge. The required representation can either be provided by a pretrained MPNN, or be learned from scratch using only structural information. Beyond the data-driven design of molecular fingerprints, the versatility of our approach is demonstrated by enabling the selection of representative entries in chemical databases, the automatic construction of coarse-grained force fields, as well as the identification of reaction coordinates
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