This paper derives the distribution of the largest principal angle between two subspaces sampled (independently) uniformly from the Grassmanian manifold of $p$-dimensional subspaces in $\mathbb{R}^d$, for $2p<d$. Is a bound known for the case that the two subspaces may have different dimensions? To phrase it more precisely, since principal angles become problematic there, I'm looking for a high probability lower bound on the smallest non-zero singular value of $U^TV$ where $U\in\mathbb{R}^{d\times k}$ and $V\in\mathbb{R}^{d\times m}$ are distributed according to the respective Haar measures, and the bound would preferably be tight especially in the regime that $d$ is significantly larger than $m$, and $k\leq m$. Ideally, the bound would be exact, i.e., non-asymptotic.

Equivalently, I'm looking for a lower bound on the eigenvalues of some $k$ by $k$ principal submatrix of $\Phi$, where $\Phi\in\mathbb{R}^{d\times d}$ is a uniform $m$-dimensional orthogonal projection.




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