reviews: 
Comments: Authors propose a neural alignment approach via hierarchical domain adaptation (HDA) in which they first align neural observations across Day 0 and Day K activity, followed by aligning their semantic latent activity. They show that HDA can perform alignment across various days of recordings better than baseline approaches.
Strengths: Authors show that HDA can improve behavior decoding performance on Day K activity over baseline methods including single-session models of neural activity, e.g., CEBRA and vanilla LSTM, and other alignment approaches such as CycleGAN. Authors also provide a theoretical foundation based on Lyapunov theory for their 2-stage alignment approach.
Weaknesses: - Major points:
  - It seems like the $E_\gamma$ is trained both on source and target domain data, rather than first training on Day 0 and aligning on Day K activity. However, I find this strategy counterintuitive for a neural alignment task. Shouldn’t authors first train their dynamic network $E_\gamma$ and decoder network $C_\eta$ on the source data initially? If the authors train their network every day from scratch with the new day’s data and source data, what is the advantage of the proposed framework over training completely a new model on new day’s data (as shown in Table 1 columns HDA and retrain)? The only advantage I can think of is the possibility of achieving better behavior decoding performance on Day K activity when the model is trained with extra data from Day 0, which does not seem to be the case. I understand that NoMAD also requires some information about Day 0 data to perform alignment such as latent factor distributions of Day 0 data, but such distributions can be computed across whole Day 0 data and stored to be used during alignment (which would not affect the alignment stage as latent factors for Day 0 data is obtained through frozen network during alignment). Even though this is not the case for ADAN, authors state that having a percentage of the Day 0 data for alignment (such as 50%, see https://github.com/limblab/adversarial_BCI/blob/main/ADAN_aligner.ipynb) is enough for the alignment stage after Day 0 training. I suspect the authors could have a similar strategy for their approach as they do not have any Day 0 training stage. I believe that this is particularly important since a BMI can be trained offline using a large dataset, but needing to have access to this large dataset during the alignment stage could limit the applicability of the decoder alignment. If the authors can, I think it is crucial to perform an ablation study on the amount of Day 0 data required for the alignment stage. 
  - It seems like NoMAD performance does not change across different days, which contradicts the findings shown in the NoMAD paper. Beyond that, I am concerned about NoMAD’s performance on Day 0, which is basically LFADS as no alignment takes place for Day 0 and it has shown great promise across multiple neural datasets (see https://eval.ai/web/challenges/challenge-page/1256/leaderboard/3184). I am surprised that LFADS, i.e., NoMAD Day 0, is performing worse than vanilla **unidirectional** LSTMs. I appreciate that the authors implemented NoMAD themselves as the public repository is not available, but authors can compare against ADAN, whose repository is publicly available, and verify if ADAN will follow a performance pattern. 
  - In line with my previous comment on LFADS, CEBRA performance on Day 0 also seems contradictory with prior findings as it again performs worse than vanilla LSTM. What positive/negative sampling strategy did the authors use while training CEBRA models? 

- Minor points:
  - NDT and STNDT do not aim to build representations that are generalized (unlike claimed in L125-126) across sessions and animals, as they are trained for individual sessions.
  - Regarding the claim about ‘independence of decomposed subspaces’ for VAEs in L272-273, are the authors explicitly learning covariances between different dimensions of latent factors and pushing them towards zero with KL divergence, or treating each latent dimension separately as commonly done in VAEs? If latter, I suspect the model is essentially learning disentangled latent variables, rather, the model tries to push individual latent dimension variances to 1, and learned nothing explicitly about their covariances.
Recommendation: 5
Reviewer confidence: 4
Soundness/correctness: 2
Clarity: 3