We have $X=[\mathbf{x}_1,\mathbf{x}_2...,\mathbf{x}_n]\in\mathbb{R}^{d\times n}$, $H=[\mathbf{h}_1,\mathbf{h}_2...,\mathbf{h}_n] \in\mathbb{R}^{d\times n}$, and $d<n$. $H$ has rank $r\leq d$ and $X$ has rank $d$.

Assume we have $\|H\|_F\leq C$ and $\|\mathbf{x}_i\|_2\leq R$, where $\|\cdot\|_F$ and $\|\cdot\|_2$ are the Frobenius norm and vectro 2-norm respectively.

What do we know about the upper bound of Frobenius inner product $\left<H,X\right>_F$ of $H$ and $X$?

Hint: I know by Cauchy-Schwarz inequality $\left<H,X\right>_F < \sqrt nCR$. The equality will not hold because we can not guarantee $\mathbf{x}_i = k\mathbf{h}_i, \forall i$. This is because $H$ is rank deficient. I think there is a tighter upper bound for $<H,X>_F$ in terms of singular values of $X$ and rank $r$ of $H$.

I have also posted the question here.