For any random $m \times n$ matrix $X$ and $m \times k$ matrix $Y$,
$$ E[\| X^T Y \|] \leq E[ \|X\|^2 ]^{1/2} E[ \|Y\|^2 ]^{1/2}, $$ where $E[\| X^T Y \|]$ can be interpreted as the absolute value of inner product, $|\langle X,Y \rangle|$, and for a $m \times n$ matrix $A$, $\|A\| = \left(\sum_{i,j}a_{ij}^2\right)^{1/2} = trace(A^T A)^{1/2}$.
I don't how can I view $E[\| X^T Y \|]$ as $|\langle X,Y \rangle|$. Could anyone explain this for me?
The inner product between two matrices $A$ and $B$ with matching sizes is usually defined as $\langle A, B \rangle = \text{tr}(A^{\top}B)$.
The expression that you have on the left-hand side can be rewritten as follows
$$ \| X^{\top} Y \| = \text{tr}^{1/2}(Y^{\top} X X^{\top}Y) = \text{tr}^{1/2}((YY^\top)^{\top} X X^{\top}). $$ On the other hand, the inner product between $X$ and $Y$ reads as $$ \langle X, Y \rangle = \text{tr}(Y^\top X). $$
The two expressions are of the same order and hence the first one can be interpreted as the second one.
