straightforward to adopt the idea of tangent space projection ([*i.e.*]{}, [Eq. ]{}) to our situation. Consider a natural [pseudometric]{} on ${\mathcal{M}}$ $$\label{eqn::metric}
d_{{\mathsf{F}}} \left(\nu_1, \nu_2\right):= \sup_{f\in {\mathsf{F}}} {\left\lVert\int f\ {\,\mathrm{d}}\nu_1 - f\ {\,\mathrm{d}}\nu_2 \right\rVert},\qquad \nu_{1,2} \in  {\mathcal{M}}.$$ Recall that ${\mathsf{F}}$ is a collection of test functions and ${\left\lVert\cdot\right\rVert}$ is some suitable norm associated with functions in ${\mathsf{F}}$. In [Section \[sec::example\]]{}, we will choose $${\mathsf{F}}= \left\{f = xx^{\dagger}\right\}$$ be a singleton and the norm ${\left\lVert\cdot\right\rVert}$ is chosen as Hilbert-Schmidt norm. An equivalent choice is that $${\mathsf{F}}= \Bigl\{\,f(x) = {\left\langlex, {{\boldsymbol{O}}}x\right\rangle} \;\big\vert \;
\text{Hermitian matrix } {{\boldsymbol{O}}}\text{ satisfies } {\lVert{{\boldsymbol{O}}}\rVert}_{{HS}} \le
1\Bigr\},$$ and the norm is simply the absolute value. The equivalence of these two choices is proved in Lemma \[lemma::choice\_equivalent\_F\] in \[sec:equivalence\]. In fact, from the perspective of quantum mechanics, this equivalence is natural, since finding a good approximation of density matrix $\int f\ {\,\mathrm{d}}\nu \equiv {\mathbb{E}}_{\nu}\bigl[xx^{\dagger}\bigr]$ is equivalent to finding a good approximation of all observations as ${\left\langle{{\boldsymbol{O}}}\right\rangle}_{\text{avg}} := {\mathbb{E}}_{\nu} \bigl[ {\left\langlex, {{\boldsymbol{O}}}x\right\rangle}\bigr] = \operatorname{tr}\left( {\mathbb{E}}_{\nu} \bigl[ {{\boldsymbol{O}}}xx^{\dagger}\bigr]\right)
= {\left\langle{{\boldsymbol{O}}}, {\mathbb{E}}_{\nu}\bigl[xx^{\dagger}\bigr]\right\rangle}_{{HS}}$ where observable ${{\boldsymbol{O}}}$ is a Hermitian matrix.

The tangent space projection of $\partial_t \mu_t = {\mathcal{A}}_{t} \mu_t$ to the tangent space ${\mathcal{T}}_{\mu_{{LR},t}} {\mathcal{M}}_r$ is then given by $$\label{eqn::sdlr_minimization}
\begin{split}
  