9 Kernel methods for surrogate modeling

Santin, Gabriele; Haasdonk, Bernard

doi:10.1515/9783110498967-009

Cited by 18 publications

(43 citation statements)

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“…The algorithm is terminated when a maximal number of nodes are selected, or when either the maximal value of the squared standard deviation, or the maximal valued of the residual fall below the tolerance τ := 10 −12 . This ensures a certain stability in the algorithm (see [26]). As GBFs we use the diffusion and the variational spline kernels (see Example 1), with parameters that are defined in each experiment.…”

Section: Methodsmentioning

confidence: 95%

“…Despite its simplicity, the method has been proven to be asymptotically optimal (see [25,35]), i.e., greedily selected nodes provide the same rate of decay of P W (IM) N as optimally chosen ones. Moreover, the algorithm has a very efficient implementation (see [20,26]). Indeed, an efficient formula is available to update P W (IM)…”

Section: P-greedy For Variance Minimizationmentioning

confidence: 99%

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Kernel-Based Models for Influence Maximization on Graphs based on Gaussian Process Variance Minimization

Cuomo¹,

Erb²,

Santin³

2021

Preprint

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The inference of novel knowledge, the discovery of hidden patterns, and the uncovering of insights from large amounts of data from a multitude of sources make Data Science (DS) to an art rather than just a mere scientific discipline. The study and design of mathematical models able to analyze information represents a central research topic in DS. In this work, we introduce and investigate a novel model for influence maximization (IM) on graphs using ideas from kernel-based approximation, Gaussian process regression, and the minimization of a corresponding variance term. Datadriven approaches can be applied to determine proper kernels for this IM model and machine learning methodologies are adopted to tune the model parameters. Compared to stochastic models in this field that rely on costly Monte-Carlo simulations, our model allows for a simple and cost-efficient update strategy to compute optimal influencing nodes on a graph. In several numerical experiments, we show the properties and benefits of this new model.

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Section: Methodsmentioning

confidence: 95%

Section: P-greedy For Variance Minimizationmentioning

confidence: 99%

Kernel-Based Models for Influence Maximization on Graphs based on Gaussian Process Variance Minimization

Cuomo¹,

Erb²,

Santin³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Proof. Let ĥ ∈ H be given by (27). Then it follows that ĥ(X) = Y − W α, ĥ(0) = 0 and D ĥ(0) = 0, i.e.…”

Section: Definition Existence and Optimality Of The Approximantmentioning

confidence: 99%

Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem

Haasdonk,

Hamzi,

Santin

et al. 2020

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For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold.Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system.In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics are quantified.We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy.

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“…In the framework of data based simulation and engineering, high-dimensional scattered data approximation and machine learning techniques are of utmost importance [30]. They construct purely data-based models which can capture the input-output behavior of complex systems through the knowledge of a finite number of measurements.…”

Section: Introductionmentioning

confidence: 99%

Universality and Optimality of Structured Deep Kernel Networks

Wenzel¹,

Santin²,

Haasdonk³

2021

Preprint

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Kernel based methods yield approximation models that are flexible, efficient and powerful. In particular, they utilize fixed feature maps of the data, being often associated to strong analytical results that prove their accuracy. On the other hand, the recent success of machine learning methods has been driven by deep neural networks (NNs). They achieve a significant accuracy on very high-dimensional data, in that they are able to learn also efficient data representations or data-based feature maps.In this paper, we leverage a recent deep kernel representer theorem to connect the two approaches and understand their interplay. In particular, we show that the use of special types of kernels yield models reminiscent of neural networks that are founded in the same theoretical framework of classical kernel methods, while enjoying many computational properties of deep neural networks. Especially the introduced Structured Deep Kernel Networks (SDKNs) can be viewed as neural networks with optimizable activation functions obeying a representer theorem. Analytic properties show their universal approximation properties in different asymptotic regimes of unbounded number of centers, width and depth. Especially in the case of unbounded depth, the constructions is asymptotically better than corresponding constructions for ReLU neural networks, which is made possible by the flexibility of kernel approximation.

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9 Kernel methods for surrogate modeling

Cited by 18 publications

References 0 publications

Kernel-Based Models for Influence Maximization on Graphs based on Gaussian Process Variance Minimization

Kernel-Based Models for Influence Maximization on Graphs based on Gaussian Process Variance Minimization

Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem

Universality and Optimality of Structured Deep Kernel Networks

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