This work is joint with Yoichiro Mori, Alexandra Jilkine, AFM (Stan)
Maree, Ben Vanderlei, and William Holmes.
I will discuss recent mathematical developments motivated by the
multivariate spline theory that demonstrate surprising connections
between these (and some other) seemingly unrelated subjects in algebra,
analysis and combinatorics.
In the last few years, a considerable progress was made on the more difficult local and non-asymptotic regimes. In the non-asymptotic regime, the dimensions of $H$ are fixed rather than grow to infinity. In the local regime, one zooms in on a small part of the spectrum of $H$ until one sees individual eigenvalues. The location of the eigenvalue nearest zero determines the invertibility properties of $H$. This essentially determines whether the matrix $H$ is well conditioned, which is a matter of importance in numerical analysis.
Examples of recent developments include the proofs that a random matrix $H$ with independent entries (whether symmetric or not) is singular with an exponentially small probability, that the condition number of $H$ is linear in the dimension, and that the eigenstructure of $H$ is delocalized and unstructured -- the eigenvectors are spread out and their coefficients are highly incommensurate.
We will see some examples of heuristics, results, and problems of the non-asymptotic random matrix theory in the local regime. Applications and problems in related areas will be discussed, in particular for covariance estimation in statistics.