1
$\begingroup$

Let $P$ be a rank $k$ uniformly randomly oriented projection matrix in ${\mathbb R}^d$ -- this is constructed as $R^T(RR^T)^{-1}R$ where $R$ is a $k\times d, k<d$ random matrix with i.i.d. 0-mean Gaussian entries with variance 1/d. Further, let T be a rank $r$ fixed deterministic projection matrix in ${\mathbb R}^d$, where $r\in\{1,...,d\}$.

Are the following true? (where $\lambda_{\max}(\cdot)$ denotes largest singular value)

  • $Pr\{\lambda_{\max}(TP)> t\}\le Pr\{\lambda_{\max}(TR^TR)> t\}$

  • $Pr\{\text{Trace}(TP)>t\}\le Pr\{\text{Trace}(TR^TR)> t\}$

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.