Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient training algorithms. In this work, we integrate concepts from pullback Riemannian geometry and generative models to propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning: score-based pullback Riemannian geometry. Focusing on unimodal distributions as a first step, we propose a score-based Riemannian structure with closed-form geodesics that pass through the data probability density. With this structure, we construct a Riemannian autoencoder (RAE) with error bounds for discovering the correct data manifold dimension. This framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training. Through numerical experiments on diverse datasets, including image data, we demonstrate that the proposed framework produces high-quality geodesics passing through the data support, reliably estimates the intrinsic dimension of the data manifold, and provides a global chart of the manifold. To the best of our knowledge, this is the first scalable framework for extracting the complete geometry of the data manifold.

In this work, we provide a mathematical proof that diffusion models encode data manifolds by approximating their normal bundles. Based on this observation we propose a novel method for extracting the intrinsic dimension of the data manifold from a trained diffusion model. Our insights are based on the fact that a diffusion model approximates the score function i.e. the gradient of the log density of a noise-corrupted version of the target distribution for varying levels of corruption. We prove that as the level of corruption decreases, the score function points towards the manifold, as this direction becomes the direction of maximal likelihood increase. Therefore, at low noise levels, the diffusion model provides us with an approximation of the manifold’s normal bundle, allowing for an estimation of the manifold’s intrinsic dimension. To the best of our knowledge our method is the first estimator of intrinsic dimension based on diffusion models and it outperforms well established estimators in controlled experiments on both Euclidean and image data.

Meta-learning models transfer the knowledge acquired from previous tasks to quickly learn new ones. They are trained on benchmarks with a fixed number of data points per task. This number is usually arbitrary and it is unknown how it affects performance at testing. Since labelling of data is expensive, finding the optimal allocation of labels across training tasks may reduce costs. Given a fixed budget of labels, should we use a small number of highly labelled tasks, or many tasks with few labels each? Should we allocate more labels to some tasks and less to others? We show that: 1) If tasks are homogeneous, there is a uniform optimal allocation, whereby all tasks get the same amount of data; 2) At fixed budget, there is a trade-off between number of tasks and number of data points per task, with a unique solution for the optimum; 3) When trained separately, harder task should get more data, at the cost of a smaller number of tasks; 4) When training on a mixture of easy and hard tasks, more data should be allocated to easy tasks. Interestingly, Neuroscience experiments have shown that human visual skills also transfer better from easy tasks. We prove these results mathematically on mixed linear regression, and we show empirically that the same results hold for few-shot image classification on CIFAR-FS and mini-ImageNet. Our results provide guidance for allocating labels across tasks when collecting data for meta-learning.