Cross product kernels for fuzzy set similarity

Guevara, Javier; Hirata, Roberto; Canu, Stéphane

doi:10.1109/fuzz-ieee.2017.8015459

Cited by 7 publications

(6 citation statements)

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“…Incorporating non-vectorial fuzzy data into a normed vectorial space such as kernel Hilbert space (RKHS) has given birth to some innovating methods [40,41], which use a positive definite kernel function specified on fuzzy sets. The kernel proposed in [42] measures the resemblance between fuzzy sets in RKHS. It has been demonstrated in this approach that such a kernel is a special case of the convolution kernel.…”

Section: Positive-definite Kernel Functions On Fuzzy Setsmentioning

confidence: 99%

Fuzzy neural network with support vector-based learning for classification and regression

Khodabandelou¹,

Ebadzadeh²

2019

Soft Comput

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Fuzzy Neural Network (FNN) and Support Vector Machine (SVM) are two prominent and powerful learning models broadly used for classification and regression. FNN has a significant local representation and human reasoning advantage. However, the drawback of such a network is that the focal point of the learning algorithms (e.g., backpropagation) is minimizing empirical risk. In contrary to FNN, SVM emphasizes simultaneously on minimizing empirical and expected risks, which theoretically leads to an excellent generalization performance power. Nevertheless, this depends on determining an adequate kernel function which is consistent with the data properties. Some approaches have been proposed to take advantage of one to deal with disadvantage of the other by combining SVM kernel and FFN. In this paper, we show that a Takagi-Sugeno-Kang (TSK)-type-based fuzzy neural network is, in fact, equivalent to an SVM with an adaptive kernel based on fuzzy rules generated in this FNN. Consequently, it is possible to learn the last layer of the FNN using the concepts of SVM and thus taking the advantage of SVM in generalization. In fact, on the one side, the proposed method is an SVM with an adaptive kernel based on fuzzy rules and on the other side, it is a TSK-FNN with SVM-based learning. Therefore, the FNN with SVM-based learning has the benefit of SVM which is minimizing both learning and testing errors. Besides, such FNN does not have the disadvantages of the SVM which is finding an appropriate kernel and training the kernel parameters. As a matter of fact, the defined kernel in FNN is an adaptive kernel based on data characteristics, which is derived from the fuzzy rules generated by the FNN itself. As a result, it is expected that the results of the proposed method for classification and regression outperform the results of SVM with the conventional kernels and fuzzy neural network trained in the conventional way. This claim is confirmed with the results obtained in this paper.

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Section: Positive-definite Kernel Functions On Fuzzy Setsmentioning

confidence: 99%

Fuzzy neural network with support vector-based learning for classification and regression

Khodabandelou¹,

Ebadzadeh²

2019

Soft Comput

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“…The cross product kernel on fuzzy sets was presented by Guevara et al [2017]. That kernel always would be positive definite if k 1 and k 2 are positive definite.…”

Section: Kernelsmentioning

confidence: 99%

“…It can be shown that k × is indeed a kind of convolution kernel (Haussler [1999]), and that under some assumptions it embeds probability distributions into RKHS. This kernel was successfully used in supervised classification on attribute noisy datasets, where it was shown that the kernel is resistant to injected random noise over the values of the predictors (see reference Guevara et al [2017] for the experiments).…”

Section: Kernelsmentioning

confidence: 99%

Kernels on fuzzy sets: an overview

Guevara¹,

Hirata²,

Canu³

2019

Preprint

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This paper introduces the concept of kernels on fuzzy sets as a similarity measure for [0, 1]-valued functions, a.k.a. membership functions of fuzzy sets. We defined the following classes of kernels: the cross product, the intersection, the non-singleton and the distance-based kernels on fuzzy sets. Applicability of those kernels are on machine learning and data science tasks where uncertainty in data has an ontic or epistemistic interpretation.

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“…Kernel functions have been designed for strings [Watkins, 1999, Lodhi et al, 2002 or more generally for sequences [Király and Oberhauser, 2019], sets [Haussler, 1999, Gärtner et al, 2002, rankings [Jiao and Vert, 2016], fuzzy domains [Guevara et al, 2017] and graphs [Borgwardt et al, 2020], which renders them broadly applicable. Their extension to the space of probability measures [Berlinet andThomas-Agnan, 2004, Smola et al, 2007] allows to represent distributions in a reproducing kernel Hilbert space (RKHS) by the so-called mean embedding.…”

Section: Introductionmentioning

confidence: 99%

Nyström $M$-Hilbert-Schmidt Independence Criterion

Kalinke¹,

Szabó²

2023

Preprint

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Kernel techniques are among the most popular and powerful approaches of data science. Among the key features that make kernels ubiquitous are (i) the number of domains they have been designed for, (ii) the Hilbert structure of the function class associated to kernels facilitating their statistical analysis, and (iii) their ability to represent probability distributions without loss of information. These properties give rise to the immense success of Hilbert-Schmidt independence criterion (HSIC) which is able to capture joint independence of random variables under mild conditions, and permits closed-form estimators with quadratic computational complexity (w.r.t. the sample size). In order to alleviate the quadratic computational bottleneck in large-scale applications, multiple HSIC approximations have been proposed, however these estimators are restricted to M " 2 random variables, do not extend naturally to the M ě 2 case, and lack theoretical guarantees. In this work, we propose an alternative Nyström-based HSIC estimator which handles the M ě 2 case, prove its consistency, and demonstrate its applicability in multiple contexts, including synthetic examples, dependency testing of media annotations, and causal discovery.

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Cross product kernels for fuzzy set similarity

Cited by 7 publications

References 10 publications

Fuzzy neural network with support vector-based learning for classification and regression

Fuzzy neural network with support vector-based learning for classification and regression

Kernels on fuzzy sets: an overview

Nyström $M$-Hilbert-Schmidt Independence Criterion

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