Embedding the group algebra into a division ring has proven to be a powerful tool for detecting structural properties of the group, especially in relation to its homology. In this talk, we will show how division rings can be used to identify residual properties of groups, one-ended groups, and coherent groups. We will place special emphasis on the class of free-by-cyclic groups to provide a clear, explicit exposition.

Recent years have witnessed a surge of interest in deep learning methods to tackle inverse problems arising in various domains such as medical imaging, remote sensing, and the arts and humanities. This talk offers an overview of recent advances in the foundations and applications of deep learning for inverse problems, with a focus on model-based deep learning methods. Concretely, this talk will overview our work relating to theoretical advances in the area of mode-based learning, including learning guarantees; algorithmic advances in model-based learning; and, finally it will showcase a portfolio of emerging signal & image processing challenges that benefit from model based learning, including image separation / deconvolution challenges arising in the arts and humanities.

Bio:

Miguel Rodrigues is a Professor of Information Theory and Processing at University College London; he leads the Information, Inference and Machine Learning Lab at UCL, and he has also been the founder and director of the master programme in Integrated Machine Learning Systems at UCL. He has also been the UCL Turing University Lead and a Turing Fellow with the Alan Turing Institute — the UK National Institute of Data Science and Artificial Intelligence.

He held various appointments with various institutions worldwide including Cambridge University, Princeton University, Duke University, and the University of Porto, Portugal. He obtained the undergraduate degree in Electrical and Computer Engineering from the Faculty of Engineering of the University of Porto, Portugal and the PhD degree in Electronic and Electrical Engineering from University College London.

Dr. Rodrigues's research lies in the general areas of information theory, information processing, and machine learning. His most relevant contributions have ranged from the information-theoretic analysis and design of communications systems, information-theoretic security, information-theoretic analysis and design of sensing systems, and the information-theoretic foundations of machine learning.

He serves or has served as Editor of IEEE BITS, Editor of the IEEE Transactions on Information Theory, and Lead Guest Editor of various Special Issues of the IEEE Journal on Selected Topics in Signal Processing, Information and Inference, and Foundations and Trends in Signal Processing.

Dr. Rodrigues has been the recipient of various prizes and awards including the Prize for Merit from the University of Porto, the Prize Engenheiro Cristian Spratley, the Prize Engenheiro Antonio de Almeida, fellowships from the Portuguese Foundation for Science and Technology, and fellowships from the Foundation Calouste Gulbenkian. Dr. Rodrigues research on information-theoretic security has also attracted the IEEE Communications and Information Theory Societies Joint Paper Award 2011.  

He has also been elevated to Fellow of the Institute of Electronics and Electrical Engineers (IEEE) for his contributions to the ‘multi-modal data processing and reliable and secure communications.’

Attention-based models, such as Transformer, excel across various tasks but lack a comprehensive theoretical understanding, especially regarding token-wise sparsity and internal linear representations. To address this gap, we introduce the single-location regression task, where only one token in a sequence determines the output, and its position is a latent random variable, retrievable via a linear projection of the input. To solve this task, we propose a dedicated predictor, which turns out to be a simplified version of a non-linear self-attention layer. We study its theoretical properties, by showing its asymptotic Bayes optimality and analyzing its training dynamics. In particular, despite the non-convex nature of the problem, the predictor effectively learns the underlying structure. This work highlights the capacity of attention mechanisms to handle sparse token information and internal linear structures.

This is a joint work with Pierre Marion, Gérard Biau and Raphaël Berthier

Cost-sensitive loss functions are crucial in many real-world prediction problems, where different types of errors are penalized differently; for example, in medical diagnosis, a false negative prediction can lead to worse consequences than a false positive prediction. However, traditional learning theory has mostly focused on the symmetric zero-one loss, letting cost-sensitive losses largely unaddressed. In this work, we extend the celebrated theory of boosting to incorporate both cost-sensitive and multi-objective losses. Cost-sensitive losses assign costs to the entries of a confusion matrix, and are used to control the sum of prediction errors accounting for the cost of each error type. Multi-objective losses, on the other hand, simultaneously track multiple cost-sensitive losses, and are useful when the goal is to satisfy several criteria at once (e.g., minimizing false positives while keeping false negatives below a critical threshold). We develop a comprehensive theory of cost-sensitive and multi-objective boosting, providing a taxonomy of weak learning guarantees that distinguishes which guarantees are trivial (i.e., can always be achieved), which ones are boostable (i.e., imply strong learning), and which ones are intermediate, implying non-trivial yet not arbitrarily accurate learning. For binary classification, we establish a dichotomy: a weak learning guarantee is either trivial or boostable. In the multiclass setting, we describe a more intricate landscape of intermediate weak learning guarantees. Our characterization relies on a geometric interpretation of boosting, revealing a surprising equivalence between cost-sensitive and multi-objective losses.