Bookmarked At the Interface of Algebra and Statistics by Tai-Danae Bradley (
This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals recover classical marginal probabilities. In general, these reduced densities will have rank higher than one, and their eigenvalues and eigenvectors will contain extra information that encodes subsystem interactions governed by statistics. We decode this information, and show it is akin to conditional probability, and then investigate the extent to which the eigenvectors capture "concepts" inherent in the original joint distribution. The theory is then illustrated with an experiment that exploits these ideas. Turning to a more theoretical application, we also discuss a preliminary framework for modeling entailment and concept hierarchy in natural language, namely, by representing expressions in the language as densities. Finally, initial inspiration for this thesis comes from formal concept analysis, which finds many striking parallels with the linear algebra. The parallels are not coincidental, and a common blueprint is found in category theory. We close with an exposition on free (co)completions and how the free-forgetful adjunctions in which they arise strongly suggest that in certain categorical contexts, the "fixed points" of a morphism with its adjoint encode interesting information.
Bookmarked Explainable Artificial Intelligence and Machine Learning: A reality rooted perspective by Frank Emmert-Streib, Olli Yli-Harja, Matthias Dehmer (
We are used to the availability of big data generated in nearly all fields of science as a consequence of technological progress. However, the analysis of such data possess vast challenges. One of these relates to the explainability of artificial intelligence (AI) or machine learning methods. Currently, many of such methods are non-transparent with respect to their working mechanism and for this reason are called black box models, most notably deep learning methods. However, it has been realized that this constitutes severe problems for a number of fields including the health sciences and criminal justice and arguments have been brought forward in favor of an explainable AI. In this paper, we do not assume the usual perspective presenting explainable AI as it should be, but rather we provide a discussion what explainable AI can be. The difference is that we do not present wishful thinking but reality grounded properties in relation to a scientific theory beyond physics.