Learning Harmonic Molecular Representations on Riemannian Manifold

Molecular representation learning plays a crucial role in AI-assisted drug discovery research. Encoding 3D molecular structures through Euclidean neural networks has become the prevailing method in the geometric deep learning community. However, the equivariance constraints and message passing in Euclidean space may limit the network expressive power. In this work, we propose a Harmonic Molecular Representation learning (HMR) framework, which represents a molecule using the Laplace-Beltrami eigenfunctions of its molecular surface. HMR offers a multi-resolution representation of molecular geometric and chemical features on 2D Riemannian manifold. We also introduce a harmonic message passing method to realize efficient spectral message passing over the surface manifold for better molecular encoding. Our proposed method shows comparable predictive power to current models in small molecule property prediction, and outperforms the state-of-the-art deep learning models for ligand-binding protein pocket classification and the rigid protein docking challenge, demonstrating its versatility in molecular representation learning.Comment: 25 pages including Appendi

Graph Priors, Optimal Transport, and Deep Learning in Biomedical Discovery

Recent advances in biomedical data collection allows the collection of massive datasets measuring thousands of features in thousands to millions of individual cells. This data has the potential to advance our understanding of biological mechanisms at a previously impossible resolution. However, there are few methods to understand data of this scale and type. While neural networks have made tremendous progress on supervised learning problems, there is still much work to be done in making them useful for discovery in data with more difficult to represent supervision. The flexibility and expressiveness of neural networks is sometimes a hindrance in these less supervised domains, as is the case when extracting knowledge from biomedical data. One type of prior knowledge that is more common in biological data comes in the form of geometric constraints. In this thesis, we aim to leverage this geometric knowledge to create scalable and interpretable models to understand this data. Encoding geometric priors into neural network and graph models allows us to characterize the models’ solutions as they relate to the fields of graph signal processing and optimal transport. These links allow us to understand and interpret this datatype. We divide this work into three sections. The first borrows concepts from graph signal processing to construct more interpretable and performant neural networks by constraining and structuring the architecture. The second borrows from the theory of optimal transport to perform anomaly detection and trajectory inference efficiently and with theoretical guarantees. The third examines how to compare distributions over an underlying manifold, which can be used to understand how different perturbations or conditions relate. For this we design an efficient approximation of optimal transport based on diffusion over a joint cell graph. Together, these works utilize our prior understanding of the data geometry to create more useful models of the data. We apply these methods to molecular graphs, images, single-cell sequencing, and health record data

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Learning cell representations in temporal and 3D contexts

Cell morphology and its changes under different circumstances is one of the primary ways by which we can understand biology. Computational tools for characterization and analysis, therefore, play a critical role in advancing studies involving cell morphology. In this thesis, I explored the use of representation learning and self-supervised methods to analyze nuclear texture in fluorescence imaging across different contexts and scales. To analyze the cell cycle using 2D temporal imaging data, as well as DNA damage in 3D imaging data, I employed a simple model based on the VAE-GAN architecture. Through the VAE-GAN model, I constructed manifolds in which the latent representations of the data can be grouped and clustered based on textural similarities without the need for exhaustive training annotations. I used these representations, as well as manually engineered features, to perform various analyses both at the single cell and tissue levels. The application on the cell cycle data revealed that common tasks such as cell cycle staging and cell cycle time estimation can be done even with minimal fluorescence information and user annotation. On the other hand, the texture classes derived to characterize DNA damage in 3D histology images unveiled differences between control and treated tissue regions. Lastly, by aggregating cell-level information to characterize local cell neighborhoods, interactions between DNA-damaged cells and immune cells can be quantified and some tissue microstructures can be identified. The results presented in this thesis demonstrated the utility of the representations learned through my approach in supporting biological inquiries involving temporal and 3D spatial data. The quantitative measurements computed using the presented methods have the potential to aid not only similar experiments on the cell cycle and DNA damage but also in exploratory studies in 3D histology

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Mass spectral imaging of clinical samples using deep learning

A better interpretation of tumour heterogeneity and variability is vital for the improvement of novel diagnostic techniques and personalized cancer treatments. Tumour tissue heterogeneity is characterized by biochemical heterogeneity, which can be investigated by unsupervised metabolomics. Mass Spectrometry Imaging (MSI) combined with Machine Learning techniques have generated increasing interest as analytical and diagnostic tools for the analysis of spatial molecular patterns in tissue samples. Considering the high complexity of data produced by the application of MSI, which can consist of many thousands of spectral peaks, statistical analysis and in particular machine learning and deep learning have been investigated as novel approaches to deduce the relationships between the measured molecular patterns and the local structural and biological properties of the tissues. Machine learning have historically been divided into two main categories: Supervised and Unsupervised learning. In MSI, supervised learning methods may be used to segment tissues into histologically relevant areas e.g. the classification of tissue regions in H&E (Haemotoxylin and Eosin) stained samples. Initial classification by an expert histopathologist, through visual inspection enables the development of univariate or multivariate models, based on tissue regions that have significantly up/down-regulated ions. However, complex data may result in underdetermined models, and alternative methods that can cope with high dimensionality and noisy data are required. Here, we describe, apply, and test a novel diagnostic procedure built using a combination of MSI and deep learning with the objective of delineating and identifying biochemical differences between cancerous and non-cancerous tissue in metastatic liver cancer and epithelial ovarian cancer. The workflow investigates the robustness of single (1D) to multidimensional (3D) tumour analyses and also highlights possible biomarkers which are not accessible from classical visual analysis of the H&E images. The identification of key molecular markers may provide a deeper understanding of tumour heterogeneity and potential targets for intervention.Open Acces

Statistical Data Modeling and Machine Learning with Applications

The modeling and processing of empirical data is one of the main subjects and goals of statistics. Nowadays, with the development of computer science, the extraction of useful and often hidden information and patterns from data sets of different volumes and complex data sets in warehouses has been added to these goals. New and powerful statistical techniques with machine learning (ML) and data mining paradigms have been developed. To one degree or another, all of these techniques and algorithms originate from a rigorous mathematical basis, including probability theory and mathematical statistics, operational research, mathematical analysis, numerical methods, etc. Popular ML methods, such as artificial neural networks (ANN), support vector machines (SVM), decision trees, random forest (RF), among others, have generated models that can be considered as straightforward applications of optimization theory and statistical estimation. The wide arsenal of classical statistical approaches combined with powerful ML techniques allows many challenging and practical problems to be solved. This Special Issue belongs to the section “Mathematics and Computer Science”. Its aim is to establish a brief collection of carefully selected papers presenting new and original methods, data analyses, case studies, comparative studies, and other research on the topic of statistical data modeling and ML as well as their applications. Particular attention is given, but is not limited, to theories and applications in diverse areas such as computer science, medicine, engineering, banking, education, sociology, economics, among others. The resulting palette of methods, algorithms, and applications for statistical modeling and ML presented in this Special Issue is expected to contribute to the further development of research in this area. We also believe that the new knowledge acquired here as well as the applied results are attractive and useful for young scientists, doctoral students, and researchers from various scientific specialties