I am studying the operator norm (spectral norm) of the following random matrix.
Let $ u_i \in \mathbb{R}^p , v_i \in \mathbb{R}^q $ be independent random vectors, where each $ u_i \sim \mathrm{Unif}(\mathbb{S}^{p-1}) , v_i \sim \mathrm{Unif}(\mathbb{S}^{q-1}) ,$
and all pairs $ (u_i, v_i) $ are independent. Let $ a_1, \dots, a_n > 0 $ be fixed scalars. Define the random matrix:
$$ M := \sum_{i=1}^n a_i u_i v_i^\top \in \mathbb{R}^{p \times q}. $$
I would like to estimate or upper bound the expected spectral norm:
$$ \mathbb{E}[\|M\|_2], $$
under the assumption that $ n < \min(p, q) $, i.e., the number of summands is smaller than the ambient dimensions.
My questions:
- What is a tight or optimal upper bound for $ \mathbb{E}[\|M\|_2] $?
I am particularly interested in whether the natural bound
$$ \mathbb{E}[\|M\|_2] \lesssim \left( \sum_{i=1}^n a_i^2 \right)^{1/2} $$
can be improved by a dimension-dependent normalization (e.g., divided by $ \sqrt{p} , \sqrt{q} , or \sqrt{pq} )$,
especially given that $ u_i, v_i $ are "spread out" over high-dimensional spheres.
- Are there any known applications of Chevet’s inequality or other Gaussian comparison tools to sums of rank-1 matrices with this structure?
Any references, sharper bounds, or relevant techniques (e.g., chaining, Gaussian width, matrix Bernstein) would be highly appreciated.
