Agenda

Speaker: Sabrina Guastavino, Università di Genova

Abstract:
This seminar explores the deep interplay between inverse problems and statistical learning, highlighting how this connection informs the development of learning algorithms and optimization strategies. We begin by framing both domains as shared optimization problems: estimating a function that maps inputs to outputs from noisy observations. Key to this formulation are the loss function and regularization. The loss function reflects assumptions about the noise model (e.g., Gaussian noise suggests a squared loss, while Poisson noise leads to the Kullback-Leibler (KL) divergence), while regularization incorporates prior knowledge about the solution, such as sparsity, often leading to feature selection techniques like Lasso.

We then show how this perspective informs the construction of learning formulations for regression and classification. For example, in Poisson noise settings with linear signal formation, a linearized KL divergence leads to a Poisson-reweighted Lasso, preserving the optimization structure of the Gaussian case while aligning with the data's statistical nature. Extending to classification, we explore score-based loss functions tailored to optimize a specific evaluation metric.

Finally, we examine the feature map not just as a kernel or learned representation but as a potentially physics-informed model. This perspective enables the integration of data-driven and theory-based approaches to uncover or approximate the true operators underlying the observed data.