Error on trace of quadratic forms from Frobenius error bound on central matrix

Question

$\DeclareMathOperator{\Tr}{Tr}$ If a Frobenius error on an estimate of covariance $\|\tilde{\boldsymbol{M}}-\boldsymbol{M}\|_F$ is known to be in the order of $\tilde{O}(f(n,d))$ as some function of $d,n$ where $M=\frac{1}{n}XX^T$ with $\mathbf{X} \in \mathbb{R}^{d \times n} \text {, where each column } X_i, i=1, \ldots, n$ corresponds to data samples. What is the corresponding bound on $\Tr(Z^T\boldsymbol{M}Z)-\Tr(Z^T\tilde{\boldsymbol{M}}Z)$ which is in terms of the trace where $Z$ is a matrix? ($Z$ has legal dimensions with respect to rest of expression). Just to clarify, all matrices over here also only have real-valued entries. Interested in both straight-forward bounds and tighter ones, if they exist.

Answer 1

The straightforward bound is to write

$$ \mathrm{Tr}(Z^T M Z) - \mathrm{Tr}(Z^T \tilde{M} Z) = \langle M - \tilde{M}, ZZ^T \rangle \leq \|ZZ^T\|_{\mathsf{F}} \|M - \tilde{M}\|_{\mathsf{F}}, $$ and substitute your estimate of the error $\|M - \tilde{M}\|_{\mathsf{F}}$.