Machine Learning has given rise to innovative technologies for a wide range of applications. In this talk, we take the perspective that understanding the (geometric) structure and the limitations of our data is crucial for the design of successful Machine Learning methods.

In the first part, we discuss how understanding the geometry of data can enable us to learn more efficiently. Many applications involve non-Euclidean data, such as graphs, strings or matrices. In such cases, it can be beneficial to perform an optimization or learning task in a space whose geometry aligns with that of the data. We will discuss several instances, where exploiting the non-Euclidean geometry of the data gives rise to Riemannian methods that are superior to standard Euclidean approaches.

In the second part, we consider the problem of learning with data limitations. Many of the recent Machine Learning success stories involve settings with access to large amounts of high-quality, labeled training data. However, in practice, such access is often limited. We discuss different instances of this problem, including (i) learning to control a system with little or no prior data on its dynamics and (ii) learning with systematic constraints on the data (e.g., privacy, biases).