Interpretable statistics for complex modelling: quantile and topological learning

As the complexity of our data increased exponentially in the last decades, so has our need for interpretable features. This thesis revolves around two paradigms to approach this quest for insights. In the first part we focus on parametric models, where the problem of interpretability can be seen as a “parametrization selection”. We introduce a quantile-centric parametrization and we show the advantages of our proposal in the context of regression, where it allows to bridge the gap between classical generalized linear (mixed) models and increasingly popular quantile methods. The second part of the thesis, concerned with topological learning, tackles the problem from a non-parametric perspective. As topology can be thought of as a way of characterizing data in terms of their connectivity structure, it allows to represent complex and possibly high dimensional through few features, such as the number of connected components, loops and voids. We illustrate how the emerging branch of statistics devoted to recovering topological structures in the data, Topological Data Analysis, can be exploited both for exploratory and inferential purposes with a special emphasis on kernels that preserve the topological information in the data. Finally, we show with an application how these two approaches can borrow strength from one another in the identification and description of brain activity through fMRI data from the ABIDE project

Generative Models of Biological Variations in Bulk and Single-cell RNA-seq

The explosive growth of next-generation sequencing data enhances our ability to understand biological process at an unprecedented resolution. Meanwhile organizing and utilizing this tremendous amount of data becomes a big challenge. High-throughput technology provides us a snapshot of all underlying biological activities, but this kind of extremely high-dimensional data is hard to interpret. Due to the curse of dimensionality, the measurement is sparse and far from enough to shape the actual manifold in the high-dimensional space. On the other hand, the measurements may contain structured noise such as technical or nuisance biological variation which can interfere downstream interpretation. Generative modeling is a powerful tool to make sense of the data and generate compact representations summarizing the embedded biological information. This thesis introduces three generative models that help amplifying biological signals buried in the noisy bulk and single-cell RNA-seq data. In Chapter 2, we propose a semi-supervised deconvolution framework called PLIER which can identify regulations in cell-type proportions and specific pathways that control gene expression. PLIER has inspired the development of MultiPLIER and has been used to infer context-specific genotype effects in the brain. In Chapter 3, we construct a supervised transformation named DataRemix to normalize bulk gene expression profiles in order to maximize the biological findings with respect to a variety of downstream tasks. By reweighing the contribution of hidden factors, we are able to reveal the hidden biological signals without any external dataset-specific knowledge. We apply DataRemix to the ROSMAP dataset and report the first replicable trans-eQTL effect in human brain. In Chapter 4, we focus on scRNA-seq and introduce NIFA which is an unsupervised decomposition framework that combines the desired properties of PCA, ICA and NMF. It simultaneously models uni- and multi-modal factors isolating discrete cell-type identity and continuous pathway-level variations into separate components. The work presented in Chapter 2 has been published as a journal article. The work in Chapter 3 and Chapter 4 are under submission and they are available as preprints on bioRxiv

Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.Comment: 13 figures, 35 reference