Classification of multivariate time series (MTS) has been tackled with a large variety of methodologies and applied to a wide range of scenarios. Reservoir computing (RC) provides efficient tools to generate a vectorial, fixed-size representation of the MTS that can be further processed by standard classifiers. Despite their unrivaled training speed, MTS classifiers based on a standard RC architecture fail to achieve the same accuracy of fully trainable neural networks. In this article, we introduce the reservoir model space, an unsupervised approach based on RC to learn vectorial representations of MTS. Each MTS is encoded within the parameters of a linear model trained to predict a low-dimensional embedding of the reservoir dynamics. Compared with other RC methods, our model space yields better representations and attains comparable computational performance due to an intermediate dimensionality reduction procedure. As a second contribution, we propose a modular RC framework for MTS classification, with an associated open-source Python library. The framework provides different modules to seamlessly implement advanced RC architectures. The architectures are compared with other MTS classifiers, including deep learning models and time series kernels. Results obtained on the benchmark and real-world MTS data sets show that RC classifiers are dramatically faster and, when implemented using our proposed representation, also achieve superior classification accuracy.

Despite the significant improvements that self-supervised representation learning has led to when learning from unlabeled data, no methods have been developed that explain what influences the learned representation. We address this need through our proposed approach, RELAX, which is the first approach for attribution-based explanations of representations. Our approach can also model the uncertainty in its explanations, which is essential to produce trustworthy explanations. RELAX explains representations by measuring similarities in the representation space between an input and masked out versions of itself, providing intuitive explanations that significantly outperform the gradient-based baselines. We provide theoretical interpretations of RELAX and conduct a novel analysis of feature extractors trained using supervised and unsupervised learning, providing insights into different learning strategies. Moreover, we conduct a user study to assess how well the proposed approach aligns with human intuition and show that the proposed method outperforms the baselines in both the quantitative and human evaluation studies. Finally, we illustrate the usability of RELAX in several use cases and highlight that incorporating uncertainty can be essential for providing faithful explanations, taking a crucial step towards explaining representations.

The Cauchy-Schwarz (CS) divergence was developed by Príncipe et al. in 2000. In this paper, we extend the classic CS divergence to quantify the closeness between two conditional distributions and show that the developed conditional CS divergence can be elegantly estimated by a kernel density estimator from given samples. We illustrate the advantages (e.g., rigorous faithfulness guarantee, lower computational complexity, higher statistical power, and much more flexibility in a wide range of applications) of our conditional CS divergence over previous proposals, such as the conditional Kullback-Leibler divergence and the conditional maximum mean discrepancy. We also demonstrate the compelling performance of conditional CS divergence in two machine learning tasks related to time series data and sequential inference, namely time series clustering and uncertainty-guided exploration for sequential decision making.

This paper presents the kernelized Taylor diagram, a graphical framework for visualizing similarities between data populations. The kernelized Taylor diagram builds on the widely used Taylor diagram, which is used to visualize similarities between populations. However, the Taylor diagram has several limitations such as not capturing non-linear relationships and sensitivity to outliers. To address such limitations, we propose the kernelized Taylor diagram. Our proposed kernelized Taylor diagram is capable of visualizing similarities between populations with minimal assumptions of the data distributions. The kernelized Taylor diagram relates the maximum mean discrepancy and the kernel mean embedding in a single diagram, a construction that, to the best of our knowledge, have not been devised prior to this work. We believe that the kernelized Taylor diagram can be a valuable tool in data visualization.