TL;DR
It is demonstrated that matrix multiplication can be reformulated as a series of convolutions by exploiting a special structure in the matrix using the Fast Fourier Trans-form (FFT), reducing the time complexity to O ( q 2 log q ) .
Multi-label learning (MLL) is a machine learning paradigm that aims to predict a set of labels for each instance, rather than a single class. Such tasks arise in a wide range of real-world applications and pose significant challenges, including an exponentially large output space, dependence among labels and often severe label imbalance. These challenges amplify predictive uncertainty, making reliable uncertainty quantification essential. Conformal prediction (CP) is an attractive answer: it converts model outputs into prediction regions with distribution-free, finite-sample guarantees under the sole assumption of data exchangeability. Several adaptations of CP to the multi-label setting have been proposed. Yet these vary widely in scoring constructions, output types and targeted guarantees. This review consolidates the landscape of CP adaptations for MLL. It places existing approaches under a unified framework, examining the types of outputs and guarantees they provide, where label dependencies are incorporated, and how inference cost scales with the number of labels. It provides an in-depth analysis of all approaches using common notation, identifying their key characteristics along with their practical implications and assessing their strengths and limitations. Finally, it compares approaches side-by-side, highlighting trade-offs among guarantee types, precision of regions, compactness of outputs and scalability. This article is part of the theme issue 'Advancing uncertainty quantification in AI systems'.
Evaluating the performance of Greedy K-center across a variety of metric spaces shows that mapping unlabeled instances into a predictive probability space and weighting the result by entropy often dominates the other options for active learning selection with Greedy K-center.
Multi-label classification methods based on the k-Nearest Neighbor (kNN) rule are widely used due to their simplicity and competitive performance, but their behavior under label-level noise remains insufficiently understood, especially when combined with data reduction techniques. This paper presents a comprehensive empirical study of the impact of label-level noise on multi-label kNN classification and on Multi-label Prototype Generation (MPG) methods. We formalize six label-level noise induction policies—Additive, Subtractive, Additive-Subtractive, Distribution-Aware Additive-Subtractive, Partial Uniform Multi-label, and Swap—parameterized by both the proportion of affected instances and a severity parameter. Their effect is analyzed on three representative kNN-based multi-label classifiers (BRkNN, LPkNN, and MLkNN) and five MPG strategies (MRHC, MChen, MRSP1-3) across eight benchmark datasets with varying label cardinality and imbalance, comprising an extensive experimental grid of 816,480 configurations. The results reveal that Additive and Partial Uniform noise are the most detrimental, whereas Subtractive and cardinality-preserving policies are comparatively less harmful. Moderate neighborhood sizes (around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=7$$\end{document}) provide a good trade-off between robustness and accuracy, while MLkNN is consistently the most resilient classifier under severe noise. Among MPG methods, MRSP3 emerges as the most robust reduction strategy, whereas aggressive reductions, particularly with MRHC, can amplify the negative effects of noise. The code and complete experimental results are publicly released to support reproducibility and further research.