Ancient Egypt is credited (along with Mesopotamia) for providing the oldest extant mathematical texts. Since the 19th century, when the first edition of the Rhind mathematical papyrus was published, it has held an important role in the historiography of mathematics. One of the earliest researchers in the field of ancient Egyptian sciences was Otto Neugebauer who has been a major influence on the early development of the field. While research in Egyptian mathematics initially focused on those aspects that could be linked to its possible successors in modern mathematics, research has also revealed various characteristics that could not easily be transferred into a modern equivalent. In addition, research on other sciences, like medicine and astronomy, has yielded further evidence that a limitation on those aspects that have successors in modern sciences will at best give an incomplete picture of ancient scholarship. This will be explored in a new long-term project, which is briefly sketched. In the context of this project, Egyptian mathematics is also studied. The talk will present an example from the terminology used in Egyptian mathematical texts to describe this area of knowledge and explore its epistemological consequences for our studies of ancient Egyptian mathematics and aim to situate it in its ancient context.

Randomized least squares is a promising method but not yet widely used in practice. We show an example of its use for finding low-rank canonical polyadic (CP) tensor decompositions for large sparse tensors. This involves solving a sequence of overdetermined least problems with special (Khatri-Rao product) structure.

In this work, we present an application of randomized algorithms to fitting the CP decomposition of sparse tensors, solving a significantly smaller sampled least squares problem at each iteration with probabilistic guarantees on the approximation errors. We perform sketching through leverage score sampling, crucially relying on the fact that the problem structure enable efficient sampling from overestimates of the leverage scores with much less work. We discuss what it took to make the algorithm practical, including general-purpose improvements.

Numerical results on real-world large-scale tensors show the method is faster than competing methods without sacrificing accuracy.

*This is joint work with Brett Larsen, Stanford University.