Overview of the AI/MATH area

The underlying questions within AI/Mathematics are very much the same questions as AI/MLX but the methods of investigation as well as the type of answers are quite different.  The search is for underlying mathematical concepts and theorems that can explain or help to understand the above described progress. Distilling mathematical ideas and mechanisms that are behind different successful applications lead to deeper understanding and cross-fertilizations.

Also in this area the recruitments will be instrumental in forming the final program. Further, as the mathematical foundations are very much in their infancy and it is too early to firmly fix a future direction, let us point to some possibilities.

Given a formal description of a mathematical well defined and slightly complicated function, say multiplication of 32-bit integers, it is computationally extremely difficult to find the best Boolean circuit realizing this function.

Training a network for a much more complicated and ill-defined task such as driving a car or recognizing a tumor should, in some intuitive sense, be much more difficult.  The fact that this is indeed possible cannot be denied and should be explained.  This is similar to the flying bumblebee. When standard models indicate that something that is possible in practice is difficult in theory, there is new interesting theory to be developed.

Robustness is a vital property for any system and within machine learning this applies both to the training stage and the classification. In some situations it is possible to relay on the robustness of the physical world but it is easy to envision higher security applications.  An adversarial perturbation of a picture, not noticeable to a human, should not make it possible to fool an image classification program.  In many such situations mathematical guarantees would be highly desirable, and we need to understand the possibilities and limitations.

Valiant introduced PAC-learning (Probably-Approximately-Correct) in 1984. This theoretically beautiful concept has seemingly little influence in modern developments.  One can only speculate if it should be abandoned or could be revised to have a larger influence.