\theta_{\vec{s},1}, \theta_{\vec{s},2} \right\}$. It coincides with the cell state signals of the LSTM units. Finally, a transformation action $a_t$ is obtained by randomly drawing from the softmax distribution conditioned on the current state $$\begin{aligned}
\label{eq6}
f_{\theta_a}\left( \vec{s}_t \right) := \mathrm{softmax} \left( \theta_{a} \vec{s}_{t} \right).\end{aligned}$$ Empirically in our paper, $\vec{o}_t$ is set to be a 1024-dim vector, $\vec{s}_t \in \mathbb{R}^{64}$ by considering 64-unit LSTM for our RNN. Subsequently the set of neural net function parameters $\mathcal{\theta}:= \left\{ \theta_{\vec{o}}, \theta_{\vec{s}}, \theta_{a} \right\}$ are of the following dimensions: For $\theta_{\vec{o}}$ of Eq. , its size is 4096 (or 8192 or 16384) $\times$ 1024, for layer $m=$3 (or 4 or 5). For $\theta_{\vec{s}}$ of Eq., the dimension is 64$\times$1024 for $\theta_{\vec{s},1}$ and 64$\times$64 for $\theta_{\vec{s},2}$. For $\theta_a$ of Eq., its size is 10$\times$64, since there are 10 distinct action types considered in our work as to be described soon.

![Exemplar action sequences executed by our localization policy.[]{data-label="fig_67"}](LP_action_Res.pdf "fig:")\

![Box-plot illustration of the distributions of action sequence length over different benchmark datasets.[]{data-label="fig_numOfActions"}](numOfActions_boxPlots.pdf "fig:"){width="0.69\linewidth"}\

At this moment, consider a Markov decision process or MDP [@SutBar:book98] that comes with a set of information states $\mathcal{S}$, a set of coordinate transformation actions $\mathcal{A}$, and a reward function $r$. 