SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In DM training, we note that  $\mathbf{Z}_0^{(s)}$  here are actually latent dynamics inferred via VAE in Eq. 6. Considering the limited number of source-domain latent dynamics, we wish to perform data augmentation so that the DM can adequately estimate  $p_s(\mathbf{Z})$ . Here we propose to enrich the input samples by learning the VAE objective (Eq. 6) and the diffusion objective (Eq. 8) cooperatively. Through the learning process of the VAE objective (i.e., ELBO), the optimization process with stochastic gradient descent (SGD) adds auxiliary perturbation to the original data samples  $\mathbf{Z}_0^{(s)}$  rather than pure Gaussian noise. This technique further fills the sample space of  $\mathbf{Z}_0^{(s)}$ , leading to better density estimation. Specifically, in each training iteration, conditioning on the current value of  $\phi_s$ , we infer a set of  $\mathbf{Z}_0 = h(\mathbf{X}^{(s)};\phi_s)$  and use it as the temporal  $\mathbf{Z}_0^{(s)}$  to optimize Eq. 8. The traditionally sequential approach is that we fully
Table 2: The coefficient of determination values  $(R^2\uparrow ,\mathrm{in}\%)$  and RMSE  $\downarrow$  of sequential source domain learning and cooperative source domain learning on the primate motor cortex dataset. Boldface denotes the highest score. Each experiment condition is repeated with 5 different random seeds, and their mean and standard deviation are listed.
Metric	Sequential	Cooperative
M1-M2	R²(%)	18.96 (±2.27)	20.47 (±2.71)
RMSE	7.76 (±0.40)	7.62 (±0.42)
M2-M3	R²(%)	21.73 (±2.46)	22.62 (±2.66)
RMSE	7.94 (±0.47)	7.73 (±0.50)
M2-C2	R²(%)	6.96 (±2.88)	8.57 (±2.96)
RMSE	11.85 (±0.42)	11.64 (±0.51)
Figure 6: Training loss curves under different neural sessions. (A) Neural session: M-1. (B) Neural session: M-3. (C) Neural session: C-2.
optimize VAE first, obtain an optimal  $\phi_s$ , and use it to optimize Eq. 8. Experimental results show that the former approach achieves higher density estimation and alignment performance. Figure 6 manifests the training loss curve in three source-target neural recording session pairs. We observe that despite the relatively under-fitting at the early stage, the cooperative source domain learning paradigm converges to solutions with lower losses and better fits. Table 2 manifests that our cooperative source domain learning paradigm leads to higher distribution alignment and neural decoding performance.
B Detailed derivation of maximum likelihood alignment
B.1 Relationship between KL-Divergence and DSM Loss
Under assumptions in Appendix A of [41], the KL divergence between the ground truth density  $q\left(\mathbf{Z} \mid \mathbf{X}^{(t)};\phi\right)$  and the DM marginal distribution  $p_0\left(\mathbf{Z};\boldsymbol{\theta}_s\right)$  can be derived as:
$$
\mathbb {D} _ {\mathrm {K L}} \left(q (\mathbf {Z} \mid \mathbf {X} ^ {(t)}; \phi) \| p _ {0} (\mathbf {Z}; \boldsymbol {\theta} _ {s})\right)
$$
$$
\begin{array}{l} \stackrel {(i)} {\leqslant} \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} \sim q (\cdot | \mathbf {X} ^ {(t)}; \phi)} \left[ \lambda (t) \left(\nabla_ {\mathbf {Z}} \log p _ {t} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) - s (\mathbf {Z}, t; \boldsymbol {\theta} _ {s})\right) \mathrm {d} \overline {{\mathbf {w}}} \right] + \mathbb {D} _ {\mathrm {K L}} \left(p _ {T} \left(\mathbf {Z}; \boldsymbol {\theta} _ {s}\right) \| \pi (\mathbf {Z})\right) \\ + \frac {1}{2} \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} \sim q (\cdot | \mathbf {X} ^ {(t)}; \phi)} \left[ \lambda (t) ^ {2} \| \nabla_ {\mathbf {Z}} \log p _ {t} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) - s (\mathbf {Z}, t; \boldsymbol {\theta} _ {s}) \| _ {2} ^ {2} d t \right] \\ \end{array}
$$
$$
\stackrel {(i i)} {=} \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} \sim q (\cdot | \mathbf {X} ^ {(t)}; \phi)} \left[ \lambda (t) ^ {2} \| \nabla_ {\mathbf {Z}} \log p _ {t} (\mathbf {Z}; \pmb {\theta} _ {s}) - s (\mathbf {Z}, t; \pmb {\theta} _ {s}) \| _ {2} ^ {2} \mathrm {d} t \right] + \mathbb {D} _ {\mathrm {K L}} \left(p _ {T} \left(\mathbf {Z}; \pmb {\theta} _ {s}\right) \| \pi (\mathbf {Z})\right)
$$
$$
\begin{array}{l} \stackrel {(i i i)} {=} \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} _ {0} \sim q (\cdot | \mathbf {X} ^ {(t)}; \phi), p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0})} \left[ \lambda (t) ^ {2} \left\| \nabla_ {\mathbf {Z} _ {t}} \log p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0}) - s (\mathbf {Z} _ {t}, t; \boldsymbol {\theta}) \right\| _ {2} ^ {2} \right] \\ + \mathbb {D} _ {\mathrm {K L}} \left(p _ {T} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \| \pi (\mathbf {Z})\right) \\ = \mathcal {L} _ {\mathrm {D S M}} \left(\phi , \boldsymbol {\theta} _ {s}\right) + \mathbb {D} _ {\mathrm {K L}} \left(p _ {T} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \| \pi (\mathbf {Z})\right), \tag {15} \\ \end{array}
$$
in which  $(i)$  is due to Girsanov Theorem [40], in  $(ii)$  we invoke the martingale property of Itô integrals [61], and in  $(iii)$  we use the denoising score matching (DSM) technique [29]. Thus we can draw to Eq. 11.
B.2 Upper bound of maximum likelihood alignment objective
In Section 3.3, by substituting Eq. 11 into Eq. 10, we have
$$
- \mathbb {E} _ {\mathbf {Z} \sim q (\mathbf {Z} | \mathbf {X} ^ {(t)}; \phi)} \left[ \log p _ {0} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \right] \leqslant \mathcal {L} _ {\mathrm {D S M}} \left(\phi , \boldsymbol {\theta} _ {s}\right) + \mathbb {D} _ {\mathrm {K L}} \left(p _ {T} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \| \pi (\mathbf {Z})\right) + \mathbb {H} \left(q (\mathbf {Z} | \mathbf {X} ^ {(t)}; \phi)\right). \tag {16}
$$
We note that the third term  $\mathbb{H}(\cdot)$  depends on the parameter set  $\phi$  of the probabilistic encoder. As  $q(\mathbf{Z}\mid \mathbf{X}^{(t)};\phi)\approx p_0(\mathbf{Z};\pmb {\theta}_s)$ , we have
$$
\begin{array}{l} \mathbb {H} \left(q (\mathbf {Z} \mid \mathbf {X} ^ {(t)}; \boldsymbol {\phi})\right) - \mathbb {H} \left(p _ {T} (\mathbf {Z}; \boldsymbol {\theta} _ {s})\right) \approx \int_ {T} ^ {0} \frac {\partial}{\partial t} \mathbb {H} \left(p _ {t} (\mathbf {Z}; \boldsymbol {\theta} _ {s})\right) d t (17) \\ \stackrel {(i)} {=} \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} \sim q (\mathbf {Z} | \mathbf {X} ^ {(t)}; \phi)} \left[ 2 f (\mathbf {Z}, t) ^ {\top} \nabla_ {\mathbf {Z}} \log p _ {t} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) - \lambda (t) ^ {2} \| \nabla_ {\mathbf {Z}} \log p _ {t} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \| _ {2} ^ {2} \right] d t \\ \stackrel {(i i)} {=} - \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} \sim q (\mathbf {Z} | \mathbf {X} ^ {(t)}; \boldsymbol {\phi})} \left[ 2 \nabla_ {\mathbf {Z}} \cdot \boldsymbol {f} (\mathbf {Z}, t) + \lambda (t) ^ {2} \| \nabla_ {\mathbf {Z}} \log p _ {t} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \| _ {2} ^ {2} \right] d t \\ \stackrel {(i i i)} {=} - \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} _ {0} \sim q (\cdot | \mathbf {X} ^ {(t)}; \phi), p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0})} \left[ 2 \nabla_ {\mathbf {Z} _ {t}} \cdot \boldsymbol {f} (\mathbf {Z} _ {t}, t) + \lambda (t) ^ {2} \| \nabla_ {\mathbf {Z} _ {t}} \log p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0}) \| _ {2} ^ {2} \right] d t (18) \\ \end{array}
$$
where in both (i) and (ii) we use integration by parts, and in (iii) we use denoising score matching (DSM) [29]. Putting the second term on the LHS of Eq. 17 into RHS and then substituting the third term on the RHS of Eq. 16, we have
$$
\begin{array}{l} - \mathbb {E} _ {\mathbf {Z} \sim q (\mathbf {Z} | \mathbf {X} ^ {(t)}; \phi)} \left[ \log p _ {0} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \right] \leqslant \mathbb {D} _ {\mathrm {K L}} \left(p _ {T} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \| \pi (\mathbf {Z})\right) (19) \\ - \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} _ {0} \sim q (\cdot | \mathbf {X} ^ {(t)}; \phi), p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0})} \left[ \lambda (t) ^ {2} \| \nabla_ {\mathbf {Z} _ {t}} \log p _ {0 t} \left(\mathbf {Z} _ {t} \mid \mathbf {Z} _ {0}\right) \| _ {2} ^ {2} \right] (20) \\ + \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} _ {0} \sim q (\cdot | \mathbf {X} ^ {(t)}; \phi), p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0})} \left[ \lambda (t) ^ {2} \| \nabla_ {\mathbf {Z} _ {t}} \log p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0}) - s (\mathbf {Z} _ {t}, t; \boldsymbol {\theta}) \| _ {2} ^ {2} \right] (21) \\ - \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} _ {0} \sim q (\cdot | \mathbf {X} ^ {(t)}; \phi), p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0})} \left[ - 2 \nabla_ {\mathbf {Z} _ {t}} \cdot \boldsymbol {f} (\mathbf {Z} _ {t}, t) \right]. (22) \\ \end{array}
$$
Since the transition probability  $p_{0t}(\mathbf{Z}_t \mid \mathbf{Z}_0)$  is a fixed Gaussian distribution and it is independent of the parameter set  $\phi$ , we can eliminate the term in Eq. 20 and rewrite the above Eqs as:
$$
\begin{array}{l} - \mathbb {E} _ {\mathbf {Z} \sim q (\mathbf {Z} | \mathbf {X} ^ {(t)}; \phi)} \left[ \log p _ {0} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \right] \leqslant \mathbb {D} _ {\mathrm {K L}} \left(p _ {T} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \| \pi (\mathbf {Z})\right) (23) \\ + \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} _ {0} \sim q (\cdot | \mathbf {X} ^ {(t)}; \phi), p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0})} \left[ \lambda (t) ^ {2} \| \nabla_ {\mathbf {Z} _ {t}} \log p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0}) - s (\mathbf {Z} _ {t}, t; \boldsymbol {\theta}) \| _ {2} ^ {2} \right] (24) \\ - \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} _ {0} \sim q (\cdot | \mathbf {X} ^ {(t)}; \phi), p _ {0 t} (\mathbf {Z} _ {t} | \mathbf {Z} _ {0})} \left[ - 2 \nabla_ {\mathbf {Z} _ {t}} \cdot \boldsymbol {f} \left(\mathbf {Z} _ {t}, t\right) \right]. (25) \\ \end{array}
$$
By substituting denoiser function  $\epsilon (\mathbf{Z}_t,t;\pmb {\theta})$  into score function  $s(\mathbf{Z}_t,t;\pmb {\theta})$  of Eq. 24, we have:
$$
\begin{array}{l} - \mathbb {E} _ {\mathbf {Z} \sim q (\mathbf {Z} | \mathbf {X} ^ {(t)}; \phi)} [ \log p _ {0} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) ] \leqslant \mathbb {D} _ {\mathrm {K L}} (p _ {T} (\mathbf {Z}; \boldsymbol {\theta} _ {s}) \| \pi (\mathbf {Z})) \\ + \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} _ {0} \sim q (\mathbf {Z} | \mathbf {X} ^ {(t)}; \boldsymbol {\phi}), \boldsymbol {\epsilon} \sim \mathcal {N} (0, I _ {l \times d})} \left[ w (t) ^ {2} \| \boldsymbol {\epsilon} - \boldsymbol {\epsilon} (\mathbf {Z} _ {t}, t; \boldsymbol {\theta} _ {s}) \| _ {2} ^ {2} - 2 \nabla_ {\mathbf {Z} _ {t}} \cdot \boldsymbol {f} (\mathbf {Z} _ {t}, t) \right]. \\ \end{array}
$$
C Detailed algorithm
C.1 Overall alignment loss function in ERDiff
Extracting the latter two terms from Eq. 12, we have the following main Maximum Likelihood Alignment (MLA) loss:
$$
\mathcal {L} _ {\mathrm {M L A}} (\boldsymbol {\phi}) = \mathbb {E} _ {t \sim \mathcal {U} [ 0, T ]} \mathbb {E} _ {\mathbf {Z} _ {0} \sim q \left(\mathbf {Z} \mid \mathbf {X} ^ {(t)}; \boldsymbol {\phi}\right), \boldsymbol {\epsilon} \sim \mathcal {N} \left(0, I _ {l \times d}\right)} \left[ w (t) ^ {2} \| \boldsymbol {\epsilon} - \boldsymbol {\epsilon} \left(\mathbf {Z} _ {t}, t; \boldsymbol {\theta} _ {s}\right) \| _ {2} ^ {2} - 2 \nabla_ {\mathbf {Z} _ {t}} \cdot \boldsymbol {f} \left(\mathbf {Z} _ {t}, t\right) \right]. \tag {26}
$$
Then, considering the Sinkhorn Regularizer term in Eq. 13:
$$
\mathcal {L} _ {\mathrm {S H D}} (\phi) = \min  _ {\gamma} \left\langle \boldsymbol {\gamma}, \mathbf {C} \right\rangle_ {F} + \lambda \mathbb {H} (\boldsymbol {\gamma}), \tag {27}
$$
where each value  $\mathbf{C}[i][j] = \left\| \mathbf{Z}_i^{(s)} - \mathbf{Z}_j^{(t)}\right\|_2^2$  in cost matrix  $\mathbf{C}$  denotes the squared Euclidean cost from  $\mathbf{Z}_i^{(s)}$  to  $\mathbf{Z}_j^{(t)}, \mathbb{H}(\boldsymbol{\gamma})$  computes the entropy of transport plan  $\boldsymbol{\gamma}$ , and  $\lambda$  refers to the weight of the entropy term. Then, we can find the optimal  $\phi_t$  via the minimization of the following total loss function:
$$
\phi_ {t} = \underset {\phi} {\operatorname {a r g m i n}} \left[ (1 - \alpha) \mathcal {L} _ {\mathrm {M L A}} + \alpha \mathcal {L} _ {\mathrm {S H D}} \right], \tag {28}
$$
where  $\alpha \in [0,1]$  is a trade-off parameter that weights the importance of Sinkhorn Regularizer term.
C.2 Algorithm for source domain learning of ERDiff
Algorithm 1 Source Domain Learning of ERDiff
Input: Source-domain neural observations  $\mathbf{X}^{(s)}$ ; Learning rate  $\eta$  and all other hyperparameters.
Output: Parameter set  $\phi^{(s)}$  of source-domain probabilistic encoder; Parameter set  $\psi^{(s)}$  of source-domain probabilistic decoder; Parameter set  $\theta^{(s)}$  of source-domain diffusion model.
1: Initialize  $\phi, \psi$ , and  $\theta$ ;
2: while not converge do
3:  $\mathcal{L}_{\mathrm{ELBO}} \gets$  Compute the loss function based on evidence lower bound according to Eq. 6;
4: Parameter Update:  $\phi \gets \phi -\eta \cdot \partial \mathcal{L}_{\mathrm{ELBO}} / \partial \phi$
5: Parameter Update:  $\psi \gets \psi -\eta \cdot \partial \mathcal{L}_{\mathrm{ELBO}} / \partial \psi$
6: Inference  $\mathbf{Z}_0^{(s)}\sim q(\cdot \mid \mathbf{X}^{(s)};\phi)$ , and Noise Sampling  $\epsilon \sim \mathcal{N}(0,I_{l\times d})$
7:  $\mathcal{L}_{\mathrm{DSM}} \gets$  Compute the denoising score matching loss according to Eq. 5;
8: Parameter Update:  $\pmb{\theta} \gets \pmb{\theta} - \eta \cdot \partial \mathcal{L}_{\mathrm{DSM}} / \partial \pmb{\theta}$
9: end while
10: return  $\phi, \psi$ , and  $\theta$ .
C.3 Algorithm for maximum likelihood alignment of ERDiff
Algorithm 2 Maximum Likelihood Alignment of ERDiff
Input: Target-domain neural observations  $\mathbf{X}^{(t)}$ ; Learning rate  $\eta$  and all other hyperparameters.
Output: Parameter set  $\phi^{(t)}$  of target-domain probabilistic encoder;
1: while not converge do
2: Inference  $\mathbf{Z}_0^{(t)}\sim q(\cdot \mid \mathbf{X}^{(t)};\phi)$  , and Noise Sampling  $\epsilon \sim \mathcal{N}(0,I_{l\times d})$
3:  $\mathcal{L}_{\mathrm{MLA}} \gets$  Compute the main Maximum Likelihood Alignment Loss according to Eq. 26;
4:  $\mathcal{L}_{\mathrm{SHD}}\gets$  Compute the Sinkhorn Regularizer according to Eq. 27;
5: Parameter Update:  $\phi \gets \phi -\eta \cdot \partial [(1 - \alpha)\mathcal{L}_{\mathrm{MLA}} + \alpha \mathcal{L}_{\mathrm{SHD}}] / \partial \phi$
6: end while
7: return  $\phi$ .
D Experimental results on rat hippocampal CA1 dataset
We further verify the effectiveness of ERDiff on a publicly available rat hippocampus dataset [27]. In this study, a rat navigated a  $1.6\mathrm{m}$  linear track, receiving rewards at both terminals (L&R) as depicted in Figure 4 (A). Concurrently, neural activity from the hippocampal CA1 region was captured, in which the neuron numbers across all recorded sessions are around 120. The neural spiking activities were binned into  $25\mathrm{ms}$  intervals. A single round trip by the rat from one end of the track to the opposite was categorized as one lap. We chose 5 sessions in total, and in each session, 80 laps were sampled. The 2-dimensional neural latent manifolds can be observed in Figure 7 (B) and (C). We also list the neural decoding results of the rat's position in Table 3.
(C)
Figure 7: Rat hippocampus dataset and experimental results. (A) Illustration of the experiment setting. (B) 2D visualization of inferred latents in the source domain, in which the bold lines manifest the mean location of latent states corresponding to the rat's position for two opposite directions. (C) On target domain, 2D visualization of inferred latents after alignment by ERDiff, DAF, and JSDM, respectively. We observe that ERDiff preserves the spatio-temporal structure of latent dynamics well.
Table 3: Decoding results after neural distribution alignment on the rat hippocampus dataset across sessions. Each experiment condition is repeated with 5 different random seeds, and their mean and standard deviation are listed.
Method	R2(%) ↑	RMSE ↓
Cycle-GAN	4.81 (±2.47)	7.65 (±0.34)
JSDM	-2.88 (±1.59)	7.65 (±0.23)
SASA	18.51 (±1.65)	6.98 (±0.20)
DANN	15.04 (±2.32)	7.30 (±0.26)
RDA-MMD	21.73 (±2.45)	7.22 (±0.26)
DAF	20.20 (±2.26)	7.36 (±0.28)
ERDiff	32.69(±2.19)	6.84(±0.26)
Table 4: Comparative analyses of computational cost between ERDiff and baseline methods during alignment. ERDiff has a comparable computational cost and maintains the stability of alignment.
Method	Cycle-GAN	JSDM	SASA	RDA-MMD	DAF	ERDiff
Add'l. Param	26K	0K	33K	65K	91K	28K
Add'l. Size	117KB	0KB	187KB	314KB	367KB	139KB
Align. Time	103ms	77ms	155ms	264ms	251ms	183ms
Stability	X	✓	✓	X	X	✓
E Computational cost analysis
Here we conduct time complexity analysis with respect to the batch size  $B$  for the alignment phase. The ERDiff's alignment objective function is composed of two main terms: Diffusion Noise Residual and Sinkhorn Divergence. We note that in the diffusion noise residual computation, it does not go through the entire  $T$  diffusion steps. Instead, it just samples a single time step (noise scale)  $t$  and calculates the noise residual specific to that step. Thus, the total complexity of this part takes  $\mathcal{O}(K_1*B*d)$ , in which the coefficient  $K_{1}$  relates to the inference complexity of the DM denoiser  $\epsilon (\mathbf{Z},t)$ ;  $d$  denotes the latent dimension size. For the Sinkhorn Divergence, it has to compute the distance matrix, costing  $\mathcal{O}(K_2*B^2)$ ;  $K_{2}$  is a relatively small coefficient in magnitude. By summing up, the total complexity of ERDiff is given by  $\mathcal{O}(K_1*B*d + K_2*B^2)$ . This  $\mathcal{O}(B^2)$  complexity is applicable since the non-adversarial baseline methods we compared (i.e., JSDM, and SASA) require quadratic complexities as well.
For any given target domain, ERDiff can stably align it to the source domain in a comparable overhead with baselines. In Table 4, we conduct a comparative analysis between ERDiff and baseline methods in terms of additional parameter number, additional model size, stability, and alignment time. The demonstrated alignment time corresponds to the execution time for aligning one iteration (a batch of size 64) on a MacBook Pro (2019 equipped with 8-Core Intel Core i9 and 4 GB RAM).