This talk discusses joint work with Venkat Chandrasekaran, Jose Israel Rodriguez, and Kevin Shu, where we initiate the study of Lagrangian dual sections. This theory gives rise to sufficient conditions for the "hidden convexity" of certain nonconvex optimization problems. Notable examples include spectral inverse problems and certain unbalanced Procrustes problems. As an additional bonus, when the constraint set is a compact Riemannian manifold, the Lagrangian formulation allows us to solve these problems using a numerical continuation algorithm based on Riemannian gradient descent.

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Many state-of-the-art methods in machine learning are black boxes which do not allow humans to understand how decisions are made. In a number of applications, like medicine and atmospheric science, researchers do not trust such black boxes. Explainable AI can be thought of as attempts to open the black box of neural networks, while interpretable AI focuses on creating clear boxes. Adversarial attacks are small perturbations of data that cause a neural network to misclassify the data or act in other undesirable ways. Such attacks are potentially very dangerous when applied to technology like self-driving cars. The goal of this talk is to introduce mathematicians to problems they can attack using their favorite mathematical tools. The mathematical structure of transformers, the powerhouse behind large language models like ChatGPT, will also be explained.

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