Kernels for Vector-Valued Functions: A Review

Álvarez, Mauricio A.; Rosasco, Lorenzo; Lawrence, Neil D.

doi:10.1561/9781601985590

Cited by 179 publications

(275 citation statements)

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“…Álvarez et al reviewed at length kernel methods for vector‐valued functions, focusing especially on regularization and Bayesian prospective, connecting the two points of view. They provided a large collection of kernel choices, focusing on separable kernels, sum of separable kernels and further extensions as kernels to learn divergence‐free and curl‐free vector fields.…”

Section: Multi‐output Regressionmentioning

confidence: 99%

A survey on multi‐output regression

Borchani

Varando

Bielza

et al. 2015

WIREs Data Min & Knowl

449

345

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In recent years, a plethora of approaches have been proposed to deal with the increasingly challenging task of multi-output regression. This paper provides a survey on state-of-the-art multi-output regression methods, that are categorized as problem transformation and algorithm adaptation methods. In addition, we present the mostly used performance evaluation measures, publicly available data sets for multi-output regression real-world problems, as well as open-source software frameworks.

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Section: Multi‐output Regressionmentioning

confidence: 99%

A survey on multi‐output regression

Borchani

Varando

Bielza

et al. 2015

WIREs Data Min & Knowl

449

345

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“…In contrast to e.g., [1,23], we do not model and treat each degree of freedom separately, but propose instead to address directly GP regression with vector outputs of dimension p. In odometry model correction, it makes particularly sense for pose residual where the orientation error highly influences the translation residual. We thus resort to a multi-output model where correlations between output components are explicitly represented through a shared input space [31]. The hyperparameter σ 2 (which leads to diagonal output covariance noise) is then replaced with a semidefinite positive (covariance) matrixΣ.…”

Section: Multi-output Gaussian Processesmentioning

confidence: 99%

Learning Wheel Odometry and IMU Errors for Localization

Brossard

Bonnabel

2019

2019 International Conference on Robotics and Automation (ICRA)

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Odometry techniques are key to autonomous robot navigation, since they enable self-localization in the environment. However, designing a robust odometry system is particularly challenging when camera and LiDAR are uninformative or unavailable. In this paper, we leverage recent advances in deep learning and variational inference to correct dynamical and observation models for state-space systems. The methodology trains Gaussian processes on the residual between the original model and the ground truth, and is applied on publicly available datasets for robot navigation based on two wheel encoders, a fiber optic gyro, and an Inertial Measurement Unit (IMU). We also propose to build an Extended Kalman Filter (EKF) on the learned model using wheel speed sensors and the fiber optic gyro for state propagation, and the IMU to update the estimated state. Experimental results clearly demonstrate that the (learned) corrected models and EKF are more accurate than their original counterparts.

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“…However, only scalar output is modeled in these studies. Direct use of GPR with multivariate output is possible by considering a covariance function that is a function of both the parameters and all components of the output [ Alvarez , ; Conti and O'Hagan , ] although the resulting GPR model will be computationally expensive if we are interested in predicting fine‐resolution solution. An alternative approach is to first reduce the degree of freedom of the multivariate output through principal component analysis [ Higdon et al ., ; Lawrence , ; Liu et al ., ] or wavelets [ Bayarri et al ., ; Drignei et al ., ; Marrel et al ., ].…”

Section: Introductionmentioning

confidence: 99%

Accurate and efficient prediction of fine‐resolution hydrologic and carbon dynamic simulations from coarse‐resolution models

Pau

Riley

Liu

2016

Water Resources Research

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The topography, and the biotic and abiotic parameters are typically upscaled to make watershed-scale hydrologic-biogeochemical models computationally tractable. However, upscaling procedure can produce biases when nonlinear interactions between different processes are not fully captured at coarse resolutions. Here we applied the Proper Orthogonal Decomposition Mapping Method (PODMM) to downscale the field solutions from a coarse (7 km) resolution grid to a fine (220 m) resolution grid. PODMM trains a reduced-order model (ROM) with coarse-resolution and fine-resolution solutions, here obtained using PAWS1CLM, a quasi-3-D watershed processes model that has been validated for many temperate watersheds. Subsequent fine-resolution solutions were approximated based only on coarse-resolution solutions and the ROM. The approximation errors were efficiently quantified using an error estimator. By jointly estimating correlated variables and temporally varying the ROM parameters, we further reduced the approximation errors by up to 20%. We also improved the method's robustness by constructing multiple ROMs using different set of variables, and selecting the best approximation based on the error estimator. The ROMs produced accurate downscaling of soil moisture, latent heat flux, and net primary production with O(1000) reduction in computational cost. The subgrid distributions were also nearly indistinguishable from the ones obtained using the fine-resolution model. Compared to coarseresolution solutions, biases in upscaled ROM solutions were reduced by up to 80%. This method has the potential to help address the long-standing spatial scaling problem in hydrology and enable long-time integration, parameter estimation, and stochastic uncertainty analysis while accurately representing the heterogeneities. Key Points:Reduced-order model for predicting high-resolution hydrological and biogeochemical dynamics Accurate and efficient estimates of soil moisture, latent heat flux, and net primary production Accurate reproduction of subgrid probability distributions, with computational savings of O(1000) (2016), Accurate and efficient prediction of fine-resolution hydrologic and carbon dynamic simulations from coarse-resolution models, Water Resour. Res., 52, 791-812,

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Kernels for Vector-Valued Functions: A Review

Cited by 179 publications

References 0 publications

A survey on multi‐output regression

A survey on multi‐output regression

Learning Wheel Odometry and IMU Errors for Localization

Accurate and efficient prediction of fine‐resolution hydrologic and carbon dynamic simulations from coarse‐resolution models

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