denotes the space of finite signed measures on ${\mathbb{C}}^r$, with bounded second moment. The low-rank dynamics on ${\mathcal{M}}_r$ that we shall consider is equivalent to the dynamics of a pair $\bigl({{\boldsymbol{U}}}(t), \theta_t\bigr)\in {V_r({{\mathbb{C}}^{{n}}})}\oplus {{\mathcal{M}}_{{\mathbb{C}}^r}}$. We remark that we use in the general framework finite signed measure instead of probability measure to avoid the subtleties arising from the geometry of probability measures (due to the positivity), see for example [@Ambrosio; @Lott17]. In practice, we will guarantee that the resulting dynamics yields probability measure by imposing more constraints on the low-rank approximation, see [Section \[sec::example\]]{}.

Tangent space projection
------------------------

It is well-known that the tangent space of ${V_r({{\mathbb{C}}^{{n}}})}$ at ${{\boldsymbol{U}}}\in {V_r({{\mathbb{C}}^{{n}}})}$ is given by (see for example [@Higham96 Theorem 1.2]), $$\label{eq:tangUV}
{\mathcal{T}}_{{{\boldsymbol{U}}}} {V_r({{\mathbb{C}}^{{n}}})}= \left\{ i {{\boldsymbol{G}}}{{\boldsymbol{U}}}: {{\boldsymbol{G}}}^{\dagger} = {{\boldsymbol{G}}}\text{ is a linear operator on }{{\mathbb{C}}^{{n}}}\right\}.$$ A self-contained proof is provided in \[sec::stiefel\_tangent\] for readers’ convenience. Thus for a differentiable trajectory ${{\boldsymbol{U}}}(t)\in {V_r({{\mathbb{C}}^{{n}}})}$, we have $\frac{{\mathrm{d}}}{{\mathrm{d}}t} {{\boldsymbol{U}}}(t) = i {{\boldsymbol{G}}}(t) {{\boldsymbol{U}}}(t)$ for some Hermitian matrices ${{\boldsymbol{G}}}(t)$.

[[Consider]{}]{} any differentiable trajectory $\theta_t$ that $$\label{eq:thetaeqn}
  \partial_t \theta_t = {\mathcal{A}}_{\theta,t} \theta_t,$$ where ${\mathcal{A}}_{\theta,t}: {{\mathcal{M}}_{{\mathbb{C}}^r}}\rightarrow {{\mathcal{M}}_{{\mathbb{C}}^r}}$ is some operator. The tangent space of ${\mathcal{M}}_{r}$ at $({{\boldsymbol{U}}}(t), \theta_t)$ is fully characterized by $${\mathcal{T}}_{{{\boldsymbol{U}}}(t)} {V_r({{\mathbb{C}}^{{n}}})}\oplus {\mathcal{T}}_{\theta_t} {{\mathcal{M}}_{{\mathbb{C}}^r}}.$$

It is 