Over the last two decades, Machine Learning (ML) and Deep Learning (DL) have revolutionized many domains by uncovering predictive structures from large, high-dimensional datasets. At the same time, leveraging these methods effectively depends critically on uncovering the inherent geometric and latent structures in both the data and the architectures of the models. The larger theme uniting this dissertation is that real-world data often reside on or near low-dimensional geometric structures embedded within high-dimensional spaces. By identifying such latent structures (geometric, probabilistic, or combinatorial), we can improve interpretability, robustness, and scalability of ML methods. Models leveraging these ideas, such as Graph Neural Networks (GNNs), diffusion-based embeddings, and spectral methods, are now commonplace in applications spanning bioinformatics, computer vision, and natural language processing. This dissertation advances the understanding and application of intrinsic geometric and latent structures through four complementary contributions: (1) We present Diffusion Earth Mover's Distance (Diffusion EMD) as a geometric distance between distributions; (2) We investigate the effectiveness of Centered Kernel Alignment (CKA) for comparing neural representations; (3) We introduce GraphPPD, a Bayesian posterior predictive framework providing uncertainty-aware graph inference; and (4) We study theoretically the spectral seriation algorithm and its robustness for recovering latent orderings from noisy graph data. First, we propose Diffusion EMD, a fast and scalable method for comparing high-dimensional datasets modeled as distributions on a shared data graph. By diffusing probability mass across the graph rather than relying on Euclidean space, Diffusion EMD aligns more closely with the manifold geometry of the data. It is topologically equivalent to classical EMD with geodesic ground distance under manifold assumptions, while being more efficient and differentiable. This enables accurate large-scale comparisons, such as those in single-cell biology, where it reveals patient-level structure by embedding sample distances into a global manifold. Second, we study the effectiveness of CKA, a popular similarity metric for neural representations. We show that CKA can be highly sensitive to simple transformations, such as affine shifts, especially in high-dimensional or small-sample settings. This raises concerns about its reliability and motivates the need for more robust alternatives. Third, we introduce GraphPPD, a variational Bayesian framework for graph-level prediction tasks. While most methods focus on node- or link-level uncertainty, GraphPPD captures full posterior predictive distributions over graph-level outputs. Built on embeddings from standard GNN backbones, it enables adaptive uncertainty quantification and improves calibration under distribution shifts, supporting more reliable decision-making. Fourth, we study spectral seriation, a classical method for recovering latent sequential orderings from pairwise similarity data. We analyze its performance under a general class of random graph models, showing that it consistently recovers hidden orderings even in the presence of significant noise. Under mild assumptions on the graph structure, spectral seriation yields ordering estimates with provable convergence rates. These results highlight the robustness of spectral methods in contexts such as genomics and recommender systems. Collectively, these contributions demonstrate the significance of intrinsic geometry and latent structures in modern ML. This dissertation follows a portfolio-style format, presenting four adapted and integrated research projects. While each chapter addresses a distinct problem, together they illuminate complementary facets of capturing, analyzing, and leveraging latent structure to advance both the theory and practice of machine learning.

This dissertation explores the intersection of optimal transport (OT), manifold learning, and generative modeling, with a primary focus on applications in computational biology. It introduces novel methods and approximations for OT (primal and dual formulations) and dynamic modeling, alongside techniques to extract geometric features from data. These contributions culminate in applications such as trajectory inference for single-cell observations at multiple time points and protein backbone generation in both unconditional and motif-scaffolding tasks. The first part focuses on extracting features from datasets, defining a diffusion-based approximation of geodesic distances. This distance is applied to data dimensionality reduction and as a cost function in OT. Additionally, a dynamic process is introduced, offering a multiscale dataset representation with convergence guarantees to a single point. This result facilitates applications in topological data analysis. The second part examines static OT approximations that implicitly employ geodesic distances as the ground cost. By leveraging graph-based diffusion operators, these methods approximate the primal and dual formulations of OT. Applications include efficiently computing Wasserstein distances, interpolating distributions, and constructing barycenters, with notable success in analyzing single-cell RNA-seq data and patient-derived organoids. The third part transitions to dynamic OT formulations and generative modeling. A novel trajectory inference method inspired by dynamic OT is proposed, which learns a vector field in a manifold-approximated space and decode the trajectory in the gene space. This provides interpretable insights into the evolution of cell populations through gene trends and groupings. Finally, a flow-based generative modeling approach on SE(3) space is introduced for protein backbone generation. Incorporating sequence conditioning, this method achieves high diversity and designability in generated proteins, offering promise for protein engineering applications. This thesis advances OT-based methods, manifold learning, and flow matching techniques, addressing challenges in scalability, interpretability, and biological data applicability. The presented methods open avenues for future research, including scaling OT approximations, integrating cell growth dynamics, and extending protein design to finer-grained representations like side chains and protein-protein interactions.

Technological advances have enabled us to gather and store data from various modalities such as text, audio, image or video at unprecedented scale. Deep learning is the signature tool that allows to understand and leverage such massive data collections, enabling us to engage in challenging new endeavors. The capabilities range from predictive tasks such as sentiment analysis in text, music and image classification, image segmentation or action recognition in video to generative tasks like generation of text, images, music and even entire videos. The success of deep learning in many applications is largely attributed to the ability of commonly used neural network architectures to leverage the intrinsic structure of the data. Image processing tasks, for example, gave rise to convolutional neural networks that rely on spatial organization of pixels, while time series analysis gave rise to recurrent neural networks that leverage temporal organization in their information processing via feedback loops and memory mechanisms. While these modalities largely reside in relatively well behaved and often highly regular domains like Euclidean spaces, further modalities that possess more abstract structure have recently attracted much attention. Data from social networks, search engines, small molecules or proteins is naturally represented by graphs and so-called geometric deep learning (GDL) has made great strides towards generalizing the design of structure-aware neural networks to such non-Euclidean domains. Among the most promising methods are graph neural networks (GNNs) that generalize the design of convolutional neural networks in vision to the graph domain. Recent advances in GNN design have introduced increasingly powerful methods for various applications, such as social network analysis, molecular predictive modeling or molecule generation. However, graph representation learning is limited by several fundamental challenges that originate from the central GNN paradigm of message passing, that is repeated averaging of node-level information across node neighborhoods. As a result, local node-level representations become either too similar from excessive averaging, or otherwise, the receptive fields of the models are too small such that information cannot be shared between distant nodes, creating a complex trade-off between so-called oversmoothing and underreaching. This dissertation presents a principled way of tackling these challenges by first deepening our understanding of the relevant data and identifying the structural properties that allow for effective graph representation learning. We consequently develop a theoretical framework rooted in graph signal processing that allows us to design powerful novel GNN architectures that provably leverage those properties, while alleviating common challenges. We find that hybrid models that combine existing methods together with novel GNN principles are particularly powerful. We provide theoretical guarantees that establish the expressive power of the proposed architectures and present exhaustive empirical analysis that demonstrates the efficacy of these novel architectures in various applications such as social networks, bio-chemistry and combinatorial optimization.

Les lentilles gravitationnelles se produisent lorsque le champ gravitationnel d'un objet massif dévie la trajectoire de la lumière provenant d'un objet lointain, entraînant une distorsion ou une amplification de l'image de l'objet lointain. La transformation Starlet fournit une méthode robuste pour obtenir une représentation éparse des images de galaxies, capturant efficacement leurs caractéristiques essentielles avec un minimum de données. Cette représentation réduit les besoins de stockage et de calcul, et facilite des tâches telles que le débruitage, la compression et l'extraction de caractéristiques. La distribution a priori de fer à cheval est une technique bayésienne efficace pour promouvoir la sparsité et la régularisation dans la modélisation statistique. Elle réduit de manière agressive les valeurs négligeables tout en préservant les caractéristiques importantes, ce qui la rend particulièrement utile dans les situations où la reconstruction d'une image originale à partir d'observations bruitées est difficile. Étant donné la nature mal posée de la reconstruction des images de galaxies à partir de données bruitées, l'utilisation de la distribution a priori devient cruciale pour résoudre les ambiguïtés. Les techniques utilisant une distribution a priori favorisant la sparsité ont été efficaces pour relever des défis similaires dans divers domaines. L'objectif principal de cette thèse est d'appliquer des techniques de régularisation favorisant la sparsité, en particulier la distribution a priori de fer à cheval, pour reconstruire les galaxies d'arrière-plan à partir d'images de lentilles gravitationnelles. Notre méthodologie proposée consiste à appliquer la distribution a priori de fer à cheval aux coefficients d'ondelettes des images de galaxies lentillées. En exploitant la sparsité de la représentation en ondelettes et le comportement de suppression du bruit de la distribution a priori de fer à cheval, nous obtenons des reconstructions bien régularisées qui réduisent le bruit et les artefacts tout en préservant les détails structurels. Des expériences menées sur des images simulées de galaxies lentillées montrent une erreur quadratique moyenne inférieure et une similarité structurelle plus élevée avec la distribution a priori de fer à cheval par rapport à d'autres méthodes, validant son efficacité.

Deep learning methods are booming. An explanation of this phenomenon is the rise of computing power combined with the accessibility of large data quantity. Nevertheless, several real-life applications present difficulties: the availability and quality of data can be low, data labeling can be tricky, etc. In this thesis, we examine two contexts: that of limited data and that of the CATS economic model. To overcome the difficulties encountered in these contexts, we use deep learning models for images with additional structure. First, we examine scattering networks and study their parameterized version on small datasets. In a second step, we adapt diffusion models in order to propose an alternative to agent-based models which are complex to build and to optimize. We empirically verify the feasibility of this approach by modeling the labor market of the CATS model. We first observe that the parameterized scattering networks perform well on classification datasets for small samples of data. We demonstrate that parameterized scattering networks perform better than those not parametrized, i.e. traditional scattering networks. Indeed, we find that filterbanks adapted to the datasets make it possible to improve learning. Moreover, we observe that the learned filters differ according to the datasets. We also verify the property of robustness to small smooth deformations experimentally.. Then, we confirm that diffusion models can be adapted to model the labor market of the CATS model in a deep learning approach. We verify this fact for two different neural network architectures. Moreover, we find that performance is maintained for different scenarios involving training with one or more time series from CATS, which can be derived from standard hyperparameters or perturbations thereof.

This thesis aligns with the trending globalization of artificial intelligence in healthcare. Through two real-world applications of recent machine learning approaches, our fundamental goal is to rigorously and intelligibly expose to the domain experts how artificial intelligence uses clinical multivariate time series to provide visualizations and predictions related to populations of patients in an emergency condition. Our results demonstrate that the recent dimensionality reduction tool PHATE combined with a clustering algorithm outperforms other more established methods in projecting multivariate time series in two dimensions and thus help the experts visualize sub-populations' trajectories. We also highlight traditional and conditional recurrent neural networks' proficiency in the early prognosis of ill patients. Finally, we allude to topological data analysis as a suitable solution to common problems related to data irregularities and incompleteness we inevitably face in the second case study.