Modern data analysis methods are expected to handle massive amounts of high dimensional data that are being collected in a variety of domains. The high dimensionality of such data introduces numerous challenges, typically referred to as the "curse of dimensionality", which render traditional statistical learning approaches impractical or ineffective for their analysis. To cope with these challenges, significant effort has been focused on developing geometric data analysis approaches that model and capture the intrinsic geometry of processed data, rather than directly modeling their distribution. In this course we will explore such approaches and provide an analytical study of the models and algorithms they use. We will start by considering supervised learning and distinguish classifiers that are based on geometric principles from posterior and likelihood estimation approaches. Next, we will consider the unsupervised learning task of clustering data and contrast approaches based on density estimation from ones that rely on metric spaces or graph constructions. Finally, we will consider more fundamental tasks in intrinsic representation learning, with particular focus on dimensionality reduction and manifold learning, e.g., with diffusion maps, tSNE, and PHATE. Time permitting, we will include guest talks on research areas related to the course, and discuss recent developments in graph signal processing and geometric deep learning.
This is a graduate-level 4 credit course at UdeM, available also via the ISM. It is suitable for CS, statistics, and applied math students interested in data science and machine learning.