**Thank you for your insightful question.**
**Q4**: Why does TopoNet-based TopoLogic and SMERF-based TopoLogic run faster and
have less params and flops compared with original TopoNet and SMERF?
**A4**: TopoNet and SMERF both compute lane topology directly using MLP. In contrast, TopoLogic calculates lane topology through a geometric distance module and a similarity module. This distinction highlights the structural differences between TopoLogic and both TopoNet and SMERF.
In the TopoNet and SMERF, two MLPs are used to generate $Q_{emb} \in \mathbb{R}^{N \times C}$, where $N$ is the number of query. Subsequently, $Q_{emb}$ are repeated to $Q_{emb}^{'} \in \mathbb{R}^{N \times N \times C}$. **They are then concatenated and processed through $\operatorname{MLP_3}$ to compute the lane topology**, as described by the following Equations:
$$
Q_{emb_1}, Q_{emb_2}=\operatorname{MLP_1}(Q_l^i), \operatorname{MLP_2}(Q_l^i) \in \mathbb{R}^{N \times C}
$$
$$
Q_{emb_1}^{'},Q_{emb_2}^{'}=\operatorname{Repeat}(Q_{emb_1}),\operatorname{Repeat}(Q_{emb_2}) \in \mathbb{R}^{N \times N \times C}
$$
$$
\operatorname{G_{sim}}=\operatorname{MLP_3}(\operatorname{Concat}(Q_{emb_1}^{'}, Q_{emb_2}^{'})) \in \mathbb{R}^{N \times N}
$$
For TopoLogic, we calculates lane topology through a geometric distance module and a similarity module. In similarity module, We directly compute lane topology via matrix multiplation between $Q_{emb_1}$ and the transposition of $Q_{emb_2}$, followed by applying a sigmoid activation function, as detailed in our paper:
$$
Q_{emb_1}, Q_{emb_2}=\operatorname{MLP_1}(Q_l^i), \operatorname{MLP_2}(Q_l^i) \in \mathbb{R}^{N \times C}
$$
$$
\operatorname{S}=\operatorname{matmul}(Q_{emb_1},\operatorname{transpose}(Q_{emb_2})) \in \mathbb{R}^{N \times N}
$$
$$
\operatorname{G_{sim}}=\operatorname{sigmoid}(\operatorname{S}) \in \mathbb{R}^{N \times N}\\
$$
In geometric distance module, as illustrated in Equation (4) of our paper, we utilize only two parameters, $\alpha$ and $\lambda$. Additionally, the merge process involves two parameters, $\lambda_1$ and $\lambda_2$, as shown in Equation (10) of our paper. All these parameters are scalar values, i.e., $\alpha,\lambda, \lambda_1, \lambda_2\in\mathbb{R}$.
To summarize, while TopoNet and SMERF models involve **three MLPs** with associated parameters, TopoLogic utilize only **two MLPs** in the similarity module, two parameters $\alpha$, $\lambda$ in geometric distance m odule, and $\lambda_1$ and $\lambda_2$ in merge process. **In TopoLogic, the scalar parameters in the geometric distance module are significantly fewer in number compared to the MLP$_3$ parameters in TopoNet and SMERF.** Additionally, the MLP$_3$ results in a higher computational complexity. Therefore, TopoLogic has a smaller number of parameters (#PARAMS) and lower computational complexity (FLOPS), leading to faster speeds (FPS) compared to TopoNet and SMERF.
**We hope this explanation addresses your concern, thank you again for your valuable feedback.**