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.

The goal of this talk is to consider two instances of a class of reconstruction problems that aim to recover an unknown signal from indirect measurements that are algebraic in nature. Such problems are paramount in mathematics, enjoying applications in a wide array of fields like molecular imaging, machine learning, and geo-positioning. In this talk, we will motivate and study the generic crystallographic phase retrieval problem and the orthogonal beltway problem and deduce conditions in each setting that guarantee signal recovery.

We discuss aspects of generalized matrix inverses from a "consistency-aware" perspective. We show that many standard tools in engineering and applied mathematics (e.g., the SVD) are commonly mis-applied in ways that undermine solution integrity. We then describe straightforward generalizations of these tools that remedy this situation.