EVENT DETAILS AND ABSTRACT

Title: Liberation of Random Projections
Speaker: Todd Kemp
Speaker Info: University of California, San Diego
Brief Description:
Special Note:
Abstract:

Consider two random subspaces $V,W\subseteq \mathbb{C}^d$. Then the dimension of $V \cap W$ is almost surely equal to $\max\{\dim V + \dim W - d,0\}$ (i.e. they are in {\em general position}). Depending on the defenition of ``almost sure'', this easy theorem dates back 150 years or more. A more modern form could be stated thus: let $P,Q$ be random projection matrices on $\mathbb{C}^d$, and let $U_t$ be an independent Brownian motion on the unitary group $U(d;\mathbb{C})$. Conjugating $P$ by $U_t$ performs a random rotation of the subspace $P(\mathbb{C}^d)$; the general position statement can be written as $\mathrm{Tr}[U_tPU_t^\ast \wedge Q] = \max\{\mathrm{Tr} P + \mathrm{Tr} Q -d,0\}$ a.s. for all $t>0$ (here $P\wedge Q$ is the projection onto $P(\mathbb{C}^d) \cap Q(\mathbb{C}^d)$). What happens when $d\to\infty$? While there is no unitarily invariant measure in infinite dimensions, it is still possible to make sense of the unitary Brownian motion and the trace for {\em some} projections on infinite-dimensional Hilbert spaces. However, the easy techniques for proving the general position theorem are unavailable. Instead, one can analyze the spectral measure $\mu_t$ of the operator $U_t P U_t^\ast Q$ for smoothness. The Cauchy transform $G(t,z)$ of $\mu_t$ satisfies a semilinear holomorphic PDE in the upper-half-plane: \[ \frac{\partial}{\partial t}G(t,z) = \frac{\partial}{\partial z}\left[z(z-1)G(t,z)^2+(az+b)G(t,z)\right]. \] I will discuss the analysis of the characteristics of this PDE, and prove not only the general position claim in infinite-dimensions but a very strong smoothing theorem that settles (a special case of) an important conjecture in free probability theory.

This is joint work with Benoit Collins.