Learning via variably scaled kernels

Campi, Cristina; Marchetti, Francesco; Perracchione, Emma

doi:10.1007/s10444-021-09875-6

Cited by 11 publications

(6 citation statements)

References 36 publications

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“…Work in progress consists in extending the proposed tool to other bases, e.g. splines [1] and to variably scaled kernels [2]. Moreover, it can be helpful for validating the shape parameter in RBF interpolation; refer e.g.…”

Section: Discussionmentioning

confidence: 99%

Efficient Reduced Basis Algorithm (ERBA) for kernel-based approximation

Marchetti¹,

Perracchione²

2021

Preprint

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The main purpose of this work is the one of providing an efficient scheme for constructing reduced interpolation models for kernel bases. In literature such problem is mainly addressed via the well-established knot insertion or knot removal schemes. Such iterative strategies are usually quite demanding from a computational point of view and our goal is to study an efficient implementation for data removal approaches, namely Efficient Reduced Basis Algorithm (ERBA). Focusing on kernelbased interpolation, the algorithm makes use of two iterative rules for removing data. The former, called ERBA-r, is based on classical residual evaluations. The latter, namely ERBA-p, is independent of the function values and relies on error bounds defined by the power function. In both cases, inspired by the so-called extended Rippa's algorithm, our ERBA takes advantage of a fast implementation.

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Section: Discussionmentioning

confidence: 99%

Efficient Reduced Basis Algorithm (ERBA) for kernel-based approximation

Marchetti¹,

Perracchione²

2021

Preprint

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“…Variably Scaled Kernels (VSKs) have been introduced in [16] in the context of kernel-based approximation, with the aim of overcoming instability issues. Then, they have been extended to work in a more general setting in [17], as presented in the following form. Let Λ ⊆ R ν , ν > 0 ∈ N and let κ : Ω × Ω −→ R be a continuous (strictly) positive definite kernel, where…”

Section: Positive Definite and Variably Scaled Kernelsmentioning

confidence: 99%

“…The function Ψ can be interpreted as a feature augmentation map, which adds ν coordinates (features) to the original sample. In this view, the VSK setting has been analysed in [17] as a stacking technique, which is capable of enhancing the prediction performances of classical kernel-based classifiers such as, e.g., SVMs. Letting x = (x 1 , .…”

Section: Positive Definite and Variably Scaled Kernelsmentioning

confidence: 99%

“…Here, we introduce what we call the Variably Scaled Persistence Kernels (VSPKs), which are inspired by the variably scaled kernels introduced in the context of approximation theory in [16] and employed as a feature augmentation strategy in [17,18] for kernel-based learning. Indeed, our aim is to build a bridge between classical variably scaled kernels and kernels defined in the context of persistent homology.…”

Section: Introductionmentioning

confidence: 99%

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Variably Scaled Persistence Kernels (VSPKs) for persistent homology applications

Marchi¹,

Lot²,

Marchetti³

et al. 2022

Preprint

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In recent years, various kernels have been proposed in the context of persistent homology to deal with persistence diagrams in supervised learning approaches. In this paper, we consider the idea of variably scaled kernels, for approximating functions and data, and we interpret it in the framework of persistent homology. We call them Variably Scaled Persistence Kernels (VSPKs). These new kernels are then tested in different classification experiments. The obtained results show that they can improve the performance and the efficiency of existing standard kernels.

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“…in KNN classification (see [1,16,17,40]). A different approach to generate anisotropic bases is represented by the so-called variably scaled kernels, which demonstrated effectiveness in encoding steep gradients and discontinuities [3,8,34,36].…”

Section: Introductionmentioning

confidence: 99%

Data-driven kernel designs for optimized greedy schemes: A machine learning perspective

Wenzel¹,

Marchetti²,

Perracchione³

2023

Preprint

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Thanks to their easy implementation via Radial Basis Functions (RBFs), meshfree kernel methods have been proved to be an effective tool for e.g. scattered data interpolation, PDE collocation, classification and regression tasks. Their accuracy might depend on a length scale hyperparameter, which is often tuned via cross validation schemes. Here we leverage approaches and tools from the machine learning community to introduce two-layered kernel machines, which generalize the classical RBF approaches that rely on a single hyperparameter. Indeed, the proposed learning strategy returns a kernel that is optimized not only in the Euclidean directions, but that further incorporates kernel rotations. The kernel optimization is shown to be robust by using recently improved calculations of cross validation scores. Finally, the use of greedy approaches, and specifically of the Vectorial Kernel Orthogonal Greedy Algorithm (VKOGA), allows us to construct an optimized basis that adapts to the data. Beyond a rigorous analysis on the convergence of the so-constructed two-Layered (2L)-VKOGA, its benefits are highlighted on both synthesized and real benchmark data sets.

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Learning via variably scaled kernels

Cited by 11 publications

References 36 publications

Efficient Reduced Basis Algorithm (ERBA) for kernel-based approximation

Efficient Reduced Basis Algorithm (ERBA) for kernel-based approximation

Variably Scaled Persistence Kernels (VSPKs) for persistent homology applications

Data-driven kernel designs for optimized greedy schemes: A machine learning perspective

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