SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Algorithm 2 Sampling Algorithm of EquiFM
1: Input: vector field model  $v_{\theta}$   
2:  $g_{1} \sim \mathcal{N}(0, I)$   
3: Subtract center of gravity from  $x_{1}$  in  $g_{1} = [x_{1}, h_{1}]$   
4:  $g_{0} = \mathrm{ODESolve}(g_{1}, v_{\theta}, 1, 0)$   
5: {During the ODE solving, always subtract the center of gravity from  $v_{\theta}^{x}$ }  
6: sample  $\hat{p}(x_{0}, h_{0}|g_{0})$   
7: return  $\langle x_{0}, h_{0} \rangle$
Note the  $\hat{p}(\cdot | g_0)$  stands for the procedure of transforming the continuous  $h_0$  into the specific data modality. This is, for the categorical part,  $\hat{p}(h | g_0) = \mathcal{C}(h | h_0)$  and for the integer part  $p(h | g_0) = \int_{h - \frac{1}{2}}^{h + \frac{1}{2}} \mathcal{N}(u | h_0, \sigma_0) \, \mathrm{d}u$ .
B Formal Proof of Theorems and Propositions
B.1 Invariant Probability Path: Theorem 4.1
The key properties of the equivariant flow matching model here are the invariant density modeling. For simplicity, here we omit the invariant feature, i.e.  $\mathbf{h}$ , and focus on the variable  $\mathbf{x}$ . Here we demonstrated that with an invariant prior  $p_1(\mathbf{x})$  and the equivariant vector field  $v_{\theta}(\mathbf{x},t)$ , the marginal distribution implied by the vector field of each time step,  $p_{\theta ,t}(\mathbf{x})$  is also invariant.
Proof. We are given that  $p_1(\mathbf{x})$  is invariant, and that  $v_{\theta}(\mathbf{x},t)$  is equivariant, i.e. for any rotation  $\mathbf{R}$ ,  $p_1(\mathbf{Rx}) = p_1(\mathbf{x})$ , and  $v_{\theta}(\mathbf{Rx},t) = \mathbf{R}v_{\theta}(\mathbf{x},t)$ . Our target is to prove that  $\forall t \in [0,1]$ ,  $\forall \mathbf{R}$ ,  $p_{\theta ,t}(\mathbf{Rx}) = p_{\theta ,t}(\mathbf{x})$ . Then specifically, the sampling distribution  $p_0(\mathbf{x})$  is invariant.
First recall that the way we generate the distribution  $p_{\theta, t}(\mathbf{x})$  is to exert a transformation  $\psi_{\theta, t}$  to the prior  $p_1(\mathbf{x})$ . And the definition of the vector field is  $v_{\theta}(\mathbf{x}, t) = \frac{\mathrm{d}}{\mathrm{d}t} \psi_{\theta, t}(\mathbf{x})$ .
We can derive the equivariance of  $\psi_{\theta ,t}$  by the following equations:
$$
\begin{array}{l} \psi_ {1} (\mathbf {x}) - \psi_ {\theta , t} (\mathbf {x}) = \int_ {t} ^ {1} v _ {\theta} (\mathbf {x}, t) \mathrm {d} t \\ \psi_ {1} (\mathbf {R x}) - \psi_ {\theta , t} (\mathbf {R x}) = \int_ {t} ^ {1} v _ {\theta} (\mathbf {R x}, t) \mathrm {d} t \\ = \int_ {t} ^ {1} \mathbf {R} v _ {\theta} (\mathbf {x}, t) \mathrm {d} t \\ = \mathbf {R} \left(\psi_ {1} (\mathbf {x}) - \psi_ {\theta , t} (\mathbf {x})\right) \\ \end{array}
$$
Note that  $\psi_{1}(\mathbf{R}\mathbf{x}) = \mathbf{R}\psi_{1}(\mathbf{x})$  since  $\psi_{1}(\mathbf{x}) = \mathbf{x}$ , we have  $\psi_{\theta ,t}(\mathbf{R}\mathbf{x}) = \mathbf{R}\psi_{\theta ,t}(\mathbf{x})$ . That is to say,  $\psi_{\theta ,t}$  is an equivariant transformation. Thus its inverse  $\psi_{\theta ,t}^{-1}$  is also equivariant, since  $\forall \mathbf{y} = \psi_{\theta ,t}(\mathbf{x})$ , we have  $\mathbf{R}\mathbf{y} = \mathbf{R}\psi_{\theta ,t}(\mathbf{x}) = \psi_{\theta ,t}(\mathbf{R}\mathbf{x})$ , so  $\psi_{\theta ,t}^{-1}(\mathbf{R}\mathbf{y}) = \mathbf{R}\mathbf{x} = \mathbf{R}\psi_{\theta ,t}^{-1}(\mathbf{y})$ . Also, the Jacobian matrix  $\frac{\partial\psi_{\theta,t}(\mathbf{x})}{\partial\mathbf{x}}$  is equivariant, i.e.  $\left.\frac{\partial\psi_{\theta,t}(\mathbf{u})}{\partial\mathbf{u}}\right|_{\mathbf{u} = \mathbf{R}\mathbf{x}} = \left.\mathbf{R}\frac{\partial\psi_{\theta,t}(\mathbf{u})}{\partial\mathbf{u}}\right|_{\mathbf{u} = \mathbf{x}}$ , which implies that  $\operatorname*{det}\frac{\partial\psi_{\theta,t}(\mathbf{u})}{\partial\mathbf{u}}|_{\mathbf{u} = \mathbf{R}\mathbf{x}} = \operatorname*{det}\frac{\partial\psi_{\theta,t}(\mathbf{u})}{\partial\mathbf{u}}|_{\mathbf{u} = \mathbf{x}}$ , since the det function keeps constant under any rotation.
According to the Change of Variable Theorem,
$$
p _ {\theta , t} (\mathbf {x}) = p _ {1} \left(\psi_ {\theta , t} ^ {- 1} (\mathbf {x})\right) / \left| \det  \frac {\partial \psi_ {\theta , t} (\mathbf {u})}{\partial \mathbf {u}} \mid_ {\mathbf {u} = \psi_ {\theta , t} ^ {- 1} (\mathbf {x})} \right|
$$
$$
p _ {\theta , t} (\mathbf {R x}) = p _ {1} \left(\psi_ {\theta , t} ^ {- 1} (\mathbf {R x})\right) / \left| \det  \frac {\partial \psi_ {\theta , t} (\mathbf {u})}{\partial \mathbf {u}} \mid_ {\mathbf {u} = \psi_ {\theta , t} ^ {- 1} (\mathbf {R x})} \right|
$$
Applying the above conclusions together, we have
$$
\begin{array}{l} p _ {\theta , t} (\mathbf {R x}) = p _ {1} (\mathbf {R} \psi_ {\theta , t} ^ {- 1} (\mathbf {x})) / \left| \det  \frac {\partial \psi_ {\theta , t} (\mathbf {u})}{\partial \mathbf {u}} \mid_ {\mathbf {u} = \psi_ {\theta , t} ^ {- 1} (\mathbf {x})} \right| \\ = p _ {1} \left(\psi_ {\theta , t} ^ {- 1} (\mathbf {x})\right) / \left| \det  \frac {\partial \psi_ {\theta , t} (\mathbf {u})}{\partial \mathbf {u}} \mid_ {\mathbf {u} = \psi_ {\theta , t} ^ {- 1} (\mathbf {x})} \right| = p _ {\theta , t} (\mathbf {x}). \\ \end{array}
$$
B.2 Explanation of Proposition 4.2
Note for the initial prior distribution,  $p_1(g|g_0)$ , could be the standard distribution, i.e.  $p_1(g|g_0) = \mathcal{N}(\mathbf{0},\mathbf{I})$ . In this case,  $p_1(g|g_0) = p_1(x|x_0)p_1(h|h_0)$ . And for the time step zero, if we assume the distribution is a Gaussian centralized on the  $g_0$ , i.e.,  $p_0(g|g_0) = \mathcal{N}(g_0,\sigma_{\min}\mathbf{I})$ . And in this case we also have that  $p_0(g|g_0) = p_0(x|x_0)p_0(h|h_0)$ .
Example 2: For the Gaussian probability path,
$$
p _ {t} (g \mid g _ {1}) = \mathcal {N} \left(g \mid \mu_ {t} (g _ {1}), \sigma_ {t} (g _ {1}) ^ {2} I\right) \tag {10}
$$
where  $\mu : [0,1] \times \mathbb{R}^d \to \mathbb{R}^d$  is the time-dependent mean of the Gaussian distribution, while  $\sigma : [0,1] \times \mathbb{R} \to \mathbb{R}_{>0}$  describes a time-dependent scalar standard deviation (std).
The Gaussian probability path satisfies that  $p_t(g|g_0) = p_t(x|x_0)p_t(h|h_0)$ . To better clarify, we highlight the difference between the conditional probability path and the marginal probability path with the following Remark B.1.
Remark B.1. With the conditional probability path on  $\mathbf{x}$  and  $\mathbf{h}$  being independent of each other, the marginal distribution could be correlated, i.e.  $p_t(g) \neq p_t(x)p_t(h)$
With this property, we could design different paths for modeling complex variables.
B.3 Proof of Proposition 4.4
In the Proposition 4.4, we claim that the probability path under the EOT map is  $\mathbf{SE}(3)$  invariant. The invariant property under translations is guaranteed by the training and sampling setting within Zero CoM space, and below we will focus on proving the path is invariant under rotations. The key observation is that  $\mathbf{R}^*$  might be different for different  $\mathbf{x}$ .
Specifically, for any rotation  $\mathbf{T}$  acting on the point cloud  $\mathbf{x}$  with  $N$  points, the new  $\mathbf{R}^*$  corresponding with  $\mathbf{T}\mathbf{x}$  (we denote it by  $\mathbf{R}_{\mathrm{rot}}^{*}$ ) exactly offsetting the impact of  $\mathbf{T}$ . Formally, let  $\mathbf{x}_0$  denote the target point cloud, we calculate
$$
(\pi^ {*}, \mathbf {R} ^ {*}) = \operatorname * {a r g m i n} _ {(\pi , \mathbf {R})} \| \pi (\mathbf {R x} ^ {1}, \mathbf {R x} ^ {2}, \dots , \mathbf {R x} ^ {N}) - \mathbf {x} _ {0} \| _ {2}
$$
$$
(\pi_ {\mathrm {r o t}} ^ {*}, \mathbf {R} _ {\mathrm {r o t}} ^ {*}) = \underset {(\pi , \mathbf {R})} {\operatorname {a r g m i n}} \| \pi (\mathbf {R T x} ^ {1}, \mathbf {R T x} ^ {2}, \dots , \mathbf {R T x} ^ {N}) - \mathbf {x} _ {0} \| _ {2}
$$
We claim that  $\pi_{\mathrm{rot}}^{*} = \pi^{*}, \mathbf{R}_{\mathrm{rot}}^{*} = \mathbf{R}^{*}\mathbf{T}^{-1}$ , and a strict proof follows.
Note that if  $\phi (\mathbf{R}):\mathbb{R}^{3\times 3}\to \mathbb{R}^{3\times 3}$  is a reversible map, then for any scalar function  $f(\mathbf{R})$  argmin  $f(\phi (\mathbf{R})) = \phi^{-1}(\underset {\mathbf{R}}{\operatorname{argmin}}f(\mathbf{R}))$ . Here let  $\phi (\mathbf{R}) = \mathbf{RT}$ , and  $\phi^{-1}(\mathbf{R}) = \mathbf{RT}^{-1}$ , we get
$$
\begin{array}{l} (\pi_ {\mathrm {r o t}} ^ {*}, \mathbf {R} _ {\mathrm {r o t}} ^ {*}) = \underset {(\pi , \mathbf {R})} {\operatorname {a r g m i n}} \| \pi (\mathbf {R T x} ^ {1}, \mathbf {R T x} ^ {2}, \dots , \mathbf {R T x} ^ {N}) - \mathbf {x} _ {0} \| _ {2} \\ = \operatorname * {a r g m i n} _ {(\pi , \mathbf {R})} \| \pi (\phi (\mathbf {R}) \mathbf {x} ^ {1}, \phi (\mathbf {R}) \mathbf {x} ^ {2}, \dots , \phi (\mathbf {R}) \mathbf {x} ^ {N}) - \mathbf {x} _ {0} \| _ {2} \\ = \hat {\phi} ^ {- 1} (\underset {(\pi , \mathbf {R})} {\operatorname {a r g m i n}} \| \pi (\mathbf {R x} ^ {1}, \mathbf {R x} ^ {2}, \dots , \mathbf {R x} ^ {N}) - \mathbf {x} _ {0} \| _ {2}) \\ = \hat {\phi} ^ {- 1} \left(\left(\pi^ {*}, \mathbf {R} ^ {*}\right)\right) = \left(\pi^ {*}, \phi^ {- 1} \left(\mathbf {R} ^ {*}\right)\right) = \left(\pi^ {*}, \mathbf {R} ^ {*} \mathbf {T} ^ {- 1}\right) \\ \end{array}
$$
where  $\hat{\phi}$  is an natural extension of  $\phi$  defined as  $\hat{\phi}((\pi, \mathbf{R})) = (\pi, \phi(\mathbf{R})), \forall (\pi, \mathbf{R})$ .
Now we recheck the probability path  $p_t$  in Eq. 8. Since  $p_1$  is invariant under rotations, and the transformation  $\psi_t^{\mathrm{EOT}}$  satisfies
$$
\begin{array}{l} \psi_ {t} ^ {\mathrm {E O T}} (\mathbf {x}) = \left(\sigma_ {\min } + (1 - \sigma_ {\min }) t\right) \pi^ {*} \left(\mathbf {R} ^ {*} \mathbf {x}\right) + (1 - t) \mathbf {x} _ {0} \\ \psi_ {t} ^ {\mathrm {E O T}} (\mathbf {T x}) = \left(\sigma_ {\min } + (1 - \sigma_ {\min }) t\right) \pi_ {\mathrm {r o t}} ^ {*} \left(\mathbf {R} _ {\mathrm {r o t}} ^ {*} \mathbf {T x}\right) + (1 - t) \mathbf {x} _ {0} \\ = \left(\sigma_ {\min } + \left(1 - \sigma_ {\min }\right) t\right) \pi^ {*} \left(\mathbf {R} ^ {*} \mathbf {x}\right) + (1 - t) \mathbf {x} _ {0} \\ \end{array}
$$
i.e.  $\psi_t^{\mathrm{EOT}}$  is invariant. So we conclude that  $p_t = [\psi_t^{\mathrm{EOT}}]_* p_1$  is also invariant under rotations.
B.4 Explanation of Proposition 4.6
Here we provide the informal explanation of the Proposition 4.6. The proposition states that for OT path or VP path on  $\mathbf{x}$  or  $\mathbf{h}$ , i.e.  $p_t(x \mid x_0) = \mathcal{N}(x \mid (1 - t)x_0, (\sigma_{\min} + (1 - \sigma_{\min})t)^2 I)$  or  $p_t(x \mid x_0) = \mathcal{N}(x \mid \alpha_t x_0, (1 - \alpha_t^2)I)$ , the mutual information between the marginal variable  $\mathbf{x}_t$  and  $\mathbf{h}_t$  monotonically decays following the path from time step 0 to time step 1 where  $I_0(\mathbf{x}_0, \mathbf{h}_0) > 0$ . Note that the  $I_1(\mathbf{x}_1, \mathbf{h}_1) = 0$ , as  $p_1(\mathbf{g}) = p_1(\mathbf{x})p_1(\mathbf{h})$ . Recall the definition of signal-to-noise ratio (SNR) as:
$$
\mathrm {S N R} = \frac {\mu^ {2}}{\sigma^ {2}}
$$
For OT-path,  $\mathrm{SNR} = \frac{(1 - t)^2}{t^2}$ ; and for VP-path  $\mathrm{SNR} = \frac{\alpha_t^2}{1 - \alpha_t^2}$ . The key observation is that the SNR along the probability path of both  $\mathbf{x}$  and  $\mathbf{h}$  on either path will decay monotonically. Intuitively, with  $t\to 1$ ,  $\mathbf{x}_t$  has less information of  $\mathbf{x}_0$  thus has less of  $\mathbf{h}$ .
Figure 5: The dependency relationship on the hybrid path. Note that  $0 < t_{1} < t_{2} < 1$ , and the direction of arrows indicate the flow corresponding to the target vector field  $u_{t}$  running backward (from  $t = 0$  to 1, opposite of the sampling process) on a random molecule  $\mathbf{g}$  (not in training data). This can be regarded as a process similar to that of a diffusion model, but  $\mathbf{x}$  and  $\mathbf{h}$  have conditionally independent paths.
B.5 Proof of Proposition 4.6
Proof. Firstly, we will prove the dependency relationship between the time-dependent variables in the hybrid probability path as shown in Fig. 5.
To this end, we demonstrate the relationship between the  $\mathbf{x}_0,\mathbf{h}_0,\mathbf{x}_{t_1}$ . Recall the definition of conditional independent probability path as  $p_t(\mathbf{x}|\langle \mathbf{x}_0,\mathbf{h}_0\rangle) = p_t(\mathbf{x}|\mathbf{x}_0)$ . Thus we have the distribution of random variable  $\mathbf{x}_{t_1},p_{t_1}(\mathbf{x}) = \int p_{t_1}(\mathbf{x}|\mathbf{x}_0)p(\mathbf{x}_0)\mathrm{d}\mathbf{x}_0$ . And hence, there is  $\mathbf{x}_{t_1}\perp \mathbf{h}_0\mid \mathbf{x}_0$ . Similarly, we could also derive  $\mathbf{h}_{t_1}\perp \mathbf{x}_0\mid \mathbf{h}_0$ .
Next, for any time step  $0 < t_1 < t_2 < 1$ . We will then demonstrate the dependency relationship between  $\mathbf{x}_0, \mathbf{x}_{t_1}, \mathbf{x}_{t_2}$ . We denote the target vector field which generates the marginal probability path  $p_t(\mathbf{x})$  as  $u_t(\mathbf{x})$ , then the distribution of random variable  $\mathbf{x}_{t_2}$  could be then derived as  $p_{t_2}(\mathbf{x}) = \left[\int_{t_1}^{t_2} u_t(\mathbf{x}) \mathrm{d}t\right]_* p_{t_1}(\mathbf{x}) = \left[\psi_{t_2}^u - \psi_{t_1}^u\right]_* p_{t_1}(\mathbf{x})$ , where  $\psi_t^u$  denotes the transformation corresponding with  $u_t$ . Then we have  $\mathbf{x}_{t_2} \perp \mathbf{x}_0 \mid \mathbf{x}_{t_1}$ . Similarly,  $\mathbf{h}_{t_2} \perp \mathbf{h}_0 \mid \mathbf{h}_{t_1}$  could be also derived.
With the above two conclusions, we demonstrate the dependency relationship shown in Fig. 5 holds in the hybrid probability path. Therefore, with the dependency relationship we could directly obtain that  $I(\mathbf{x}_{t_2}, \mathbf{h}_{t_2}) \leq I(\mathbf{x}_{t_1}, \mathbf{h}_{t_1})$ . If  $\alpha_t$  in VP path satisfies  $\alpha_t > 0$  when  $0 < t < 1$ , then for any combination of such OT path and VP path, we always have  $I(\mathbf{x}_t, \mathbf{h}_t) > 0$ .
C.1 Solving EOT with a variant of iterative closest point (ICP) algorithm.
Problem definition. Given a point cloud  $\mathbf{z} \in \mathbb{R}^{3 \times N}$  and its reference point cloud  $\mathbf{y} \in \mathbb{R}^{3 \times N}$ , note they are point cloud representations in Euclidean space. The objective is to find an optimal permutation matrix  $\boldsymbol{\Pi}^* \in \mathbb{R}^{N \times N}$  and a rotation matrix  $\mathbf{R}^* \in \mathbb{R}^{3 \times 3}$  that minimizes the following objective:
$$
\boldsymbol {\Pi} ^ {*}, \mathbf {R} ^ {*} = \underset {\boldsymbol {\Pi}, \mathbf {R}} {\operatorname {a r g m i n}} \| \boldsymbol {\Pi} (\mathbf {R} \mathbf {z}) ^ {\top} - \mathbf {y} ^ {\top} \| _ {2} \tag {11}
$$
We optimize the objective iteratively with a variant of the iterative closest point (ICP) algorithm, where it iteratively obtains  $\Pi$  and  $\mathbf{R}$ .
C.2 Model Architectures and Training Configurations
We use the open-source software RDKIT [21] to preprocess molecules. For QM9 we take atom types (H, C, N, O, F) and integer-valued atom charges as atomic features, while for DRUG we only use atom types.
Algorithm 3 A variant of iterative closest point (ICP) algorithm.
1: Input: a point cloud  $\mathbf{z} \in \mathbb{R}^{3 \times N}$  and it's reference point cloud  $\mathbf{y} \in \mathbb{R}^{3 \times N}$ .  
2: while  $\tau$  has not converged do  
3: Obtain permutation matrix  $\Pi = \underset{\Pi}{\mathrm{argmin}} \| \Pi (\mathbf{Rz})^{\top} - \mathbf{y}^{\top} \|_{2}$  with Jonker-Volgenant algorithm [7]  
4: Obtain rotation matrix  $\mathbf{R} = \underset{\mathbf{R}}{\mathrm{argmin}} \| \mathbf{R} (\Pi \mathbf{z})^{\top} - \mathbf{y}^{\top} \|_{2}$  with Kabsch algorithm [22]  
5:  $\tau = \| \Pi (\mathbf{Rz}^{\top})^{\top} - \mathbf{y}^{\top} \|_{2}$   
6: end while  
7: return  $\Pi, \mathbf{R}$
The vector field network is implemented with EGNNs [43] by PyTorch [34] package. We set the dimension of latent invariant features  $k$  to 1 for QM9 and 2 for DRUG, which extremely reduces the atomic feature dimension. For the training of vector field network  $v_{\theta}$ : on QM9, we train EGNNs with 9 layers and 256 hidden features with a batch size 64; and on DRUG, we train EGNNs with 4 layers and 256 hidden features, with batch size 64. The model uses SiLU activations. We train all the modules until convergence. For all the experiments, we choose the Adam optimizer [19] with a constant learning rate of  $10^{-4}$  as our default training configuration. The training on QM9 takes approximately 2000 epochs, and on DRUG takes 20 epochs.
D Number of Evaluation (NFE) analysis
We further explore the behavior of adaptive integrators during sampling with Dopri15 as an example. In Fig. 7, we show the average NFE at different time intervals. We could observe that at time intervals near 0 the NFE is much larger than at other time intervals. The underlying reason lies in the vector field of  $\mathbf{h}$  dramatically changes in these steps, which results in the frequent change of the atom type in the last period of sampling. This behavior could be due to the unsmoothness of the categorical manifold which could shed light on several future directions.
E Scalability
The proposed algorithm has a complexity of  $O(n^{2})$  for computing an OT map for a single molecule, where  $n$  is the number of nodes. To understand the computational burden for different molecule sizes, we evaluated the average computing time for OT maps with varying node numbers. The Tab. 4 below shows the burden for three datasets/tasks. We also provide the curves of iteration and time needed to solve the EOT map in Fig. 6.
(a) iterations towards atom numbers
Figure 6: Time Cost for Solving EOT Maps
(b) time towards atom numbers
Even for the Antibody-CDR data, with an average of 150 atoms, the process time is only  $18.84\mathrm{ms}$  which is acceptable in practice. Additionally, we can optimize the process further by leveraging its
Table 4: Scalability to Different Molecule Size
Average atom number	EOT mapping time per sample	EOT mapping iteration per sample
QM9	18	1.10ms	4.67
GEOM DRUG	47	1.99ms	5.89
Antibody-CDR	150	18.84ms	7.14
Figure 7: Number of Evaluation analysis of EquiFM generation process with Dopri5 integrator.
parallelizable nature. By Enabling prefetch and multiprocessing, we can minimize the computational overhead further and make it virtually inconsequential.
F More visualizations
This section presents additional visualizations of molecules generated by our EquiFM method. We include samples from two datasets, QM9 and DRUG, in Fig. 8 and Fig. ??, respectively. All examples are randomly generated without cherry-picking, but the viewing direction may affect the visibility of some geometries.
As demonstrated in the figures, our model can generate realistic molecular geometries for small and large molecules alike. However, the model occasionally generates disconnected components, which is more common when trained on the large molecule DRUG dataset, as shown in the second molecule in Fig. ?? This phenomenon is not unique to our model but is a common issue in non-autoregressive molecule generative models [54, 16]. Nevertheless, it is easily solvable by filtering out the smaller components.
We also present a qualitative assessment of controlled molecule generation by EquiFM in Fig. 9. We interpolate the conditioning parameter, polarizability, with different values of  $\alpha$ , while keeping the prior  $g_{1}$  fixed. Polarizability measures the tendency of matter to acquire an electric dipole moment when subjected to an electric field. In general, less isometric molecular geometries tend to correspond to higher  $\alpha$  values. This observation is consistent with our results in Fig. 9.
Figure 8: Molecules generated from EquiFM trained on QM9.
Figure 9: Molecules generated from conditioned version of EquiFM trained on QM9. We conduct controllable generation with interpolation among different polarizability  $\alpha$  values with the same prior  $g_{1}$ . The given  $\alpha$  values are provided at the bottom.