Metric Flow Matching for Smooth Interpolations on the Data Manifold

Matching objectives underpin the success of modern generative models and rely on constructing conditional paths that transform a source distribution into a target distribution. Despite being a fundamental building block, conditional paths have been designed principally under the assumption of Euclidean geometry, resulting in straight interpolations. However, this can be particularly restrictive for tasks such as trajectory inference, where straight paths might lie outside the data manifold, thus failing to capture the underlying dynamics giving rise to the observed marginals. In this paper, we propose Metric Flow Matching (MFM), a novel simulation-free framework for conditional flow matching where interpolants are approximate geodesics learned by minimizing the kinetic energy of a data-induced Riemannian metric. This way, the generative model matches vector fields on the data manifold, which corresponds to lower uncertainty and more meaningful interpolations. We prescribe general metrics to instantiate MFM, independent of the task, and test it on a suite of challenging problems including LiDAR navigation, unpaired image translation, and modeling cellular dynamics. We observe that MFM outperforms the Euclidean baselines, particularly achieving SOTA on single-cell trajectory prediction.

CIN++: Enhancing Topological Message Passing

Graph Neural Networks (GNNs) have demonstrated remarkable success in learning from graph-structured data. However, they face significant limitations in expressive power, struggling with long-range interactions and lacking a principled approach to modeling higher-order structures and group interactions. Cellular Isomorphism Networks (CINs) recently addressed most of these challenges with a message passing scheme based on cell complexes. Despite their advantages, CINs make use only of boundary and upper messages which do not consider a direct interaction between the rings present in the underlying complex. Accounting for these interactions might be crucial for learning representations of many real-world complex phenomena such as the dynamics of supramolecular assemblies, neural activity within the brain, and gene regulation processes. In this work, we propose CIN++, an enhancement of the topological message passing scheme introduced in CINs. Our message passing scheme accounts for the aforementioned limitations by letting the cells to receive also lower messages within each layer. By providing a more comprehensive representation of higher-order and long-range interactions, our enhanced topological message passing scheme achieves state-of-the-art results on large-scale and long-range chemistry benchmarks.

Expressiveness Remarks for Denoising Diffusion Models and Samplers

Denoising diffusion models are a class of generative models which have recently achieved state-of-the-art results across many domains. Gradual noise is added to the data using a diffusion process, which transforms the data distribution into a Gaussian. Samples from the generative model are then obtained by simulating an approximation of the time reversal of this diffusion initialized by Gaussian samples. Recent research has explored adapting diffusion models for sampling and inference tasks. In this paper, we leverage known connections to stochastic control akin to the F\"ollmer drift to extend established neural network approximation results for the F\"ollmer drift to denoising diffusion models and samplers.

Graph Neural Networks for Breast Cancer Data Integration

International initiatives such as METABRIC (Molecular Taxonomy of Breast Cancer International Consortium) have collected several multigenomic and clinical data sets to identify the undergoing molecular processes taking place throughout the evolution of various cancers. Numerous Machine Learning and statistical models have been designed and trained to analyze these types of data independently, however, the integration of such differently shaped and sourced information streams has not been extensively studied. To better integrate these data sets and generate meaningful representations that can ultimately be leveraged for cancer detection tasks could lead to giving well-suited treatments to patients. Hence, we propose a novel learning pipeline comprising three steps - the integration of cancer data modalities as graphs, followed by the application of Graph Neural Networks in an unsupervised setting to generate lower-dimensional embeddings from the combined data, and finally feeding the new representations on a cancer sub-type classification model for evaluation. The graph construction algorithms are described in-depth as METABRIC does not store relationships between the patient modalities, with a discussion of their influence over the quality of the generated embeddings. We also present the models used to generate the lower-latent space representations: Graph Neural Networks, Variational Graph Autoencoders and Deep Graph Infomax. In parallel, the pipeline is tested on a synthetic dataset to demonstrate that the characteristics of the underlying data, such as homophily levels, greatly influence the performance of the pipeline, which ranges between 51\% to 98\% accuracy on artificial data, and 13\% and 80\% on METABRIC. This project has the potential to improve cancer data understanding and encourages the transition of regular data sets to graph-shaped data.