Training of machine learning models for classification frequently yields several different solutions although the performance remains approximately the same, i.e. one observes many close-to-optimum solutions with only marginal performance differences which are, however, qualitatively well-distinguishable.This behaviour is known as the Rashomon effect and may be dedicated to the stochastic in the training process, different learning strategies or various initial settings.Hence, model explanations may become difficult and have to be related to a given configuration.Therefore, stable and plausible explanations are required based on the evaluation of the Rashomon set.Yet, the consistency of the resulting explanations remained largely unexplored so far.Here we propose to evaluate the Rashomon set qualitatively by means of a cluster analysis based on the determination of the feature importance.Feature importance of a model gives insights about the decision making process and, hence, provides an appropriate criterion to distinguish model decision realizations.Clustering of them reveal stable and plausible classification strategies and, hence, contribute to reliable explanations.

Counterfactuals are useful objects to explain decisions of machine learning classifiers.In the best case, counterfactuals can provide to derive causal inference structure realized by the model.Yet, counterfactual generation in general is known as a constrained optimization problem.In this contribution we demonstrate that counterfactuals can be determined geometric-analytically in case of prototype based classifiers.For this we only require that nearest prototype classification is based on norms induced by an inner product, which has to be applied for consistency also to evaluate the deviation between a given sample and a desired counterfactual class.* M.K. and L.R. are supported by

Machine learning classifiers adjust implicitly or explicitly the importance of the data features to solve a given classification task.This feature weighting often does not imply feature sparseness, which, however, may be important for interpretability and model evaluation.This contribution proposes how to force feature sparseness in combination with feature relevance for prototype-based classification learning to obtain reliable and interpretable classification decisions.

Developing fair classification models is a crucial aspect of machine learning research. However, unintended distortion in training data - biased data - can lead to discriminatory decisions. In this paper, we developed a workflow for detecting and mitigating bias in data using a shallow, interpretable machine learning models: the Generalized Matrix Learning Vector Quantization. We extent the approach by a relevance-based analysis to identify and reduce bias in the data. Combining similarity metric adaptation and relevance-based analysis, we can develop fair classification models that minimize the influence of bias in the data. Our results demonstrate that this method is effective in reducing bias in classification models and therefore supports fair decision-making.

We propose a bias detection and mitigating scheme for data in the context of classification tasks based on learning vector quantizers (LVQ) as classifier.For this purpose generalized LVQ endowed with an advanced matrix adaptation scheme is used for bias detection.The bias removal from data is realized applying a nullspace data projection using the adjusted matrix.The usefulness of the approach is demonstrated and illustrated in terms of two real world datasets.* M.K. is

An interpretable approach to classification learning using cross-entropy loss is the Probabilistic Learning Vector Quantizer (PLVQ) as a robust prototype-based classifier.We propose a variant of the PLVQ, that allows the integration of domain knowledge.This strategy is becoming increasingly popular as a means of developing intelligent models that can enhance performance and gain acceptance from domain experts.In this paper, we put forth the idea of incorporating externally known class relations as supplementary information.We present theoretical aspects of the model and demonstrate its capabilities through numerical experiments.

Knowledge integration into machine learning systems is a promising and successful strategy to achieve more plausible and consistent results.The plausibility is accompanied by better model interpretability due to the adjustment of the machine learning system to the domain specic requirements and restrictions.Further, informed machine learning can be seen as a particular task specic regularization of the model leading to better learning convergence and frequently also requiring a lower amount of training data.This short introduction paper addresses some recent aspects, how domain knowledge can be integrated into learning systems on dierent levels ranging from informed feature extraction to domain adjusted structure and model architecture.* M.K. is supported by the IAI-

Approximation problems, and thus regression problems, have been widely considered as machine learning problems.A popular model to tackle such tasks are radial-basis-function networks (RBFN) and variants thereof.However, due to the global approximation scheme, RBFN, when trained in a supervised manner without additional constraints, may lack local representation.To this end, we propose approaches that aim to preserve locality in terms of the regression problem by using the Neural Gas algorithm.The models are tested on different data sets and compared to the supervised RBFN approach.* R. S. is supported by

The encounter of large amounts of biological sequence data generated during the last decades and the algorithmic and hardware improvements have offered the possibility to apply machine learning techniques in bioinformatics. While the machine learning community is aware of the necessity to rigorously distinguish data transformation from data comparison and adopt reasonable combinations thereof, this awareness is often lacking in the field of comparative sequence analysis. With realization of the disadvantages of alignments for sequence comparison, some typical applications use more and more so-called alignment-free approaches. In light of this development, we present a conceptual framework for alignment-free sequence comparison, which highlights the delineation of: 1) the sequence data transformation comprising of adequate mathematical sequence coding and feature generation, from 2) the subsequent (dis-)similarity evaluation of the transformed data by means of problem-specific but mathematically consistent proximity measures. We consider coding to be an information-loss free data transformation in order to get an appropriate representation, whereas feature generation is inevitably information-lossy with the intention to extract just the task-relevant information. This distinction sheds light on the plethora of methods available and assists in identifying suitable methods in machine learning and data analysis to compare the sequences under these premises.

We present an approach for generating vectorial representations of graphs for machine learning applications based on a sensoric response principle and multiple graph kernels.The sensor perspective reduces the graph kernel computations significantly.Thus, multiple kernel (relevance) learning can be realized using the interpretable generalized matrix learning vector quantization (GMLVQ) classifier.Results obtained in small molecule classification serve as proof of concept.* K.S.B and M.K. are supported by a grant of the European Social

One-class-classification remains an important problem in machine learning, which is related to data representation and outlier detection, but different from them in several aspects. In the present contribution we propose an one-class-classifier based on a prototype vector quantization model. We modeled a corresponding cost function to account for aspects of representation learning and to appropriately evaluate the one-class classifier. The prototype-based model ensures a local representation of the target class. After this introduction, we obtain an interpretable one-class classifier model. We demonstrate the capabilities of the approach by applying the classifier to illustrative toy data examples as well as on real data in a medical context.

We present an approach to discriminate SARS-CoV-2 virus types based on their RNA sequence descriptions avoiding a sequence alignment. For that purpose, sequences are preprocessed by feature extraction and the resulting feature vectors are analyzed by prototype-based classification to remain interpretable. In particular, we propose to use variants of learning vector quantization (LVQ) based on dissimilarity measures for RNA sequence data. The respective matrix LVQ provides additional knowledge about the classification decisions like discriminant feature correlations and, additionally, can be equipped with easy to realize reject options for uncertain data. Those options provide self-controlled evidence, i.e., the model refuses to make a classification decision if the model evidence for the presented data is not sufficient. This model is first trained using a GISAID dataset with given virus types detected according to the molecular differences in coronavirus populations by phylogenetic tree clustering. In a second step, we apply the trained model to another but unlabeled SARS-CoV-2 virus dataset. For these data, we can either assign a virus type to the sequences or reject atypical samples. Those rejected sequences allow to speculate about new virus types with respect to nucleotide base mutations in the viral sequences. Moreover, this rejection analysis improves model robustness. Last but not least, the presented approach has lower computational complexity compared to methods based on (multiple) sequence alignment. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s00521-021-06018-2.

Abstract Prototype-based models like the Generalized Learning Vector Quantization (GLVQ) belong to the class of interpretable classifiers. Moreover, quantum-inspired methods get more and more into focus in machine learning due to its potential efficient computing. Further, its interesting mathematical perspectives offer new ideas for alternative learning scenarios. This paper proposes a quantum computing-inspired variant of the prototype-based GLVQ for classification learning. We start considering kernelized GLVQ with real- and complex-valued kernels and their respective feature mapping. Thereafter, we explain how quantum space ideas could be integrated into a GLVQ using quantum bit vector space in the quantum state space $${\mathcal {H}}^{n}$$ H n and show the relations to kernelized GLVQ. In particular, we explain the related feature mapping of data into the quantum state space $${\mathcal {H}}^{n}$$ H n . A key feature for this approach is that $${\mathcal {H}}^{n}$$ H n is an Hilbert space with particular inner product properties, which finally restrict the prototype adaptations to be unitary transformations. The resulting approach is denoted as Qu-GLVQ. We provide the mathematical framework and give exemplary numerical results.

In this paper we present a realization of the informationbottleneck-paradigm by means of an improved counter propagation network.It combines an unsupervised vector quantizer for data compression with a subsequent supervised learning vector quantization model.The approach is mathematically justified and yields an interpretable model for classification under the constraint of data compression, which is not longer independently learned from the classification task.* M.K., M.

Learning Vector Quantizers (LVQ) and its cost-functionbased variant called Generalized Learning Vector Quanitzation (GLVQ) are powerful, yet simple and interpretable classification models.Even though GLVQ is an effective tool for classifying vectorial data, it cannot handle raw sequence data of potentially different lengths.Usually, this problem is solved by manually engineering fixed-length features or by employing recurrent networks.Therefore, a natural idea is to incorporate recurrent units for data processing into the GLVQ network structure.The processed data can then be compared in a latent space for classification decisions.We demonstrate the ability of this approach on illustrative classification problems.* M.K. and J.R. are

Abstract In this contribution, we consider the classification of time series and similar functional data which can be represented in complex Fourier and wavelet coefficient space. We apply versions of learning vector quantization (LVQ) which are suitable for complex-valued data, based on the so-called Wirtinger calculus. It allows for the formulation of gradient-based update rules in the framework of cost-function-based generalized matrix relevance LVQ (GMLVQ). Alternatively, we consider the concatenation of real and imaginary parts of Fourier coefficients in a real-valued feature vector and the classification of time-domain representations by means of conventional GMLVQ. In addition, we consider the application of the method in combination with wavelet-space features to heartbeat classification.

Classification and decision systems in data analysis are mostly based on accuracy optimization. This criterion is only a conditional informative value if the data are imbalanced or false positive/negative decisions cause different costs. Therefore more sophisticated statistical quality measures are favored in medicine, like precision, recall etc‥ Otherwise, most classification approaches in machine learning are designed for accuracy optimization. In this paper we consider variants of learning vector quantizers (LVQs) explicitly optimizing those advanced statistical quality measures while keeping the basic intuitive ingredients of these classifiers, which are the prototype based principle and the Hebbian learning. In particular we focus in this contribution particularly to precision and recall as important measures for use in medical applications. We investigate these problems in terms of precision-recall curves as well as receiver-operating characteristic (ROC) curves well-known in statistical classification and test analysis. With the underlying more general framework, we provide a principled alternatives traditional classifiers, such that a closer connection to statistical classification analysis can be drawn.

In the present paper we investigate the analysis of functional magnetic resonance image (fMRI) data based on voxel response analysis. All voxels in local spatial area (volume) of a considered voxel form its so-called searchlight. The searchlight for a presented task is taken as a complex pattern. Task dependent discriminant analysis of voxel is then performed by assessment of the discrimination behavior of the respective searchlight pattern for a given task. Classification analysis of these patterns is usually done using linear support vector machines (linSVMs) as a machine learning approach or another statistical classifier like linear discriminant classifier. The test classification accuracy determining the task sensitivity is interpreted as the discrimination ability of the related voxel. However, frequently, the number of voxels contributing to a searchlight is much larger than the number of available pattern samples in classification learning, i.e. the dimensionality of patterns is higher than the number of samples. Therefore, the respective underlying mathematical classification problem has not an unique solution such that a certain solution obtained by the machine learning classifier contains arbitrary (random) components. For this situation, the generalization ability of the classifier may drop down. We propose in this paper another data processing approach to reduce this problem. In particular, we reformulate the classification problem within the searchlight. Doing so, we avoid the dimensionality problem: We obtain a mathematically well-defined classification problem, such that generalization ability of a trained classifier is kept high. Hence, a better stability of the task discrimination is obtained. Additionally, we propose the utilization of generalized learning vector quantizers as an alternative machine learning classifier system compared to SVMs, to improve further the stability of the classifier model due to decreased model complexity.

In the present paper we investigate the application of differentiable kernel for generalized matrix learning vector quantization as an alternative kernel-based classifier, which additionally provides classification dependent data visualization. We show that the concept of differentiable kernels allows a prototype description in the data space but equipped with the kernel metric. Moreover, using the visualization properties of the original matrix learning vector quantization we are able to optimize the class visualization by inherent visualization mapping learning also in this new kernel-metric data space.

The contribution describes the application of the Team 'Computational Intelligence Group' from the University of Applied Sciences Mittweida (Germany) to the ICMLA Face Recognition Challenge 2012. In particular we explain the data preprocessing and feature extraction, which was applied before classification learning. Further we give details about the used classification algorithm - the enhanced generalized matrix learning vector quantization model (eGMLVQ). We provide information about the results as well as observed classification properties detected by the learning algorithm.