$M(t)$ by $M_{{LR}}(t)\in \mathcal{M}_r$ for all $t$; secondly, the time-evolution $M_{{LR}}(t)$ is given by $$\label{eqn::tsp_ode}
  \frac{{\,\mathrm{d}}}{{\,\mathrm{d}}t} M_{{LR}}(t) = \operatorname*{arg\,min}_{v \in {\mathcal{T}}_{M_{{LR}}(t)} {\mathcal{M}}_r} d\bigl(v, F(t, M_{{LR}}(t))\bigr),$$ where ${\mathcal{T}}_{M_{{LR}}(t)} {\mathcal{M}}_r$ is the tangent space of $\mathcal{M}_r$ at the current location $M_{{LR}}(t)$ and $d(v_1, v_2) := {\left\lVertv_1 - v_2\right\rVert}$ is a metric on tangent space; thus the evolution is constructed as close as possible to the solution of matrix ODE by projecting $F(t, M_{{LR}}(t))$ onto the tangent space ${\mathcal{T}}_{M_{{LR}}(t)}{\mathcal{M}}_r$.

In our proposed method, we adopt this idea to the space of finite signed measures on ${{\mathbb{C}}^{{n}}}$ with bounded second moment, denoted by ${\mathcal{M}}$. The subspace ${\mathcal{M}}_r$ in this case is defined as the space of finite signed measures supported on a linear subspace of ${{\mathbb{C}}^{{n}}}$ with dimension at most $r$. Then, we hope to approximate $\mu_t$ by $\mu_{{LR},t}\in
{\mathcal{M}}_r$. The time-evolution equation of the low-rank approximation is given by $$\partial_t \mu_{{LR},t} \equiv {\mathcal{A}}_{{LR},t} \mu_{{LR},t} := \operatorname*{arg\,min}_{\nu \in {\mathcal{T}}_{\mu_{{LR},t}}{\mathcal{M}}_r}  d_{{\mathsf{F}}}\left(\nu,\ {\mathcal{A}}_{t} \mu_{{LR},t}\right),$$ where $d_{{\mathsf{F}}}$ is a [pseudometric]{} defined in [Eq. ]{} below and ${\mathcal{T}}_{\mu_{{LR},t}} {\mathcal{M}}_r$ is the tangent space of ${\mathcal{M}}_r$ at $\mu_{{LR},t}$. We will refer this method as the [stochastic dynamical [low-rank]{} approximation]{} method (or [SDLR]{} in abbreviation).

As a concrete example, let 