Representation learning and exploration of data geometries

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In the past two decades, rapid technological advances have revolutionized the availability of big high dimensional data in virtually every field of science and technology. Modern data are being produced at increasingly large volumes, quantified by numerous measurements, and often collected in many batches or samples that encode nontrivial variations (e.g., between different observation settings, collection times, or collection technologies). Such availability of raw observational data provides potential for new discoveries at a level that was never possible before. However, it also introduces numerous data analysis challenges that require a new generation of machine learning tools to enable unsupervised exploration of data and enhance the ability of domain experts to extract new knowledge and insights from massive datasets, rather than automate their ability to annotate them. The proposed research program directly ties into the growing interest in exploratory data analysis tools that extract simplified data representations for inferring or uncovering emergent patterns, structures, and dynamics. A strong emphasis in these is on providing tractable, human-interpretable, representations that make data approachable for study, hypothesis generation, and further exploration by domain experts who are not necessarily computation oriented. Concretely, this proposal will focus on constructing such representations by leveraging tools from manifold learning, graph signal processing, and the recently emerging field of geometric deep learning. While numerous data-intensive fields are relevant and in need of such data exploration tools, the proposed research is expected to have immediate and direct impact on two application fields. First, single cell data analysis introduces exciting new prospects for understanding high resolution biological heterogeneity, but also gives rise to various challenges due to batch effects, high dimenstionality, and sparsity of the collected data. Furthermore, exploratory processing of single cell data often requires separation of intrinsic data geometry from data distribution as rare subpopulations and sparse transitions between meta-stable states are often of great interest in biomedical data analysis, but they would be lost by an analysis that focuses on the main distribution modes. Second, the study of latent representations in deep neural networks has recently gained popularity in order to understand their impressive machine learning capabilities. Indeed, as cascades of linear operations and relatively-simple nonlinearities, neural networks naturally provide task-oriented internal representations given by neuron activations in hidden layers. Here, exploratory data analysis can help understand the gradual information processing mechanisms that emerge as neural nets specialize on varied tasks and shed light on the effects of data distribution on the intrinsic structure of neuron activation spaces (e.g., activation manifolds).