QLunch: Jeong-Hoon Ju

Speaker: Jeong-Hoon Ju

Title: Structured Matrix Factorization Length

Abstract: Ye and Lim (2016) proved that every (resp. a generic) complex $n \times n$ matrix can be expressed as a product of $2n+5$ (resp. $\lfloor n/2 \rfloor +1$) Toeplitz matrices. Motivated by this result, it is natural to ask the following question: what is the minimum number of Toeplitz matrices required to factor a given matrix? We generalize this question from Toeplitz structure to more general structures. In this talk, we introduce the notion of structured matrix factorization length when the set of matrices with a given structure is an affine variety. Then we define its algebro-geometric analogue, the border structured matrix factorization length. Finally, we discuss methods for deriving upper and lower bounds for these quantities. For lower bound, we suggest a method based on displacement rank; for upper bound, we suggest an approach using alternating minimization. This is joint work with Taehyeong Kim.