SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Figure 14: Performance over smaller datasets. Each datapoint is the mean ROC-AUC of 10 different splits while keeping scaffolds, with standard deviations shown as error bars. From the upper left to the lower right: Bace, BBBP, Clintox, Sider.
E Low-Data Scenario
Here we perform a more detailed experiment to illustrate how model performances change with increasing numbers of training samples. We fine-tune multiple models (GraphCL, JOAO, GraphLoG and SimGRACE) and TDL versions on 8 molecular property prediction datasets with training sets of different size. The experiment of each low-data size is run with 10 different subsets of the original training sets while keeping scaffolds. Figure 13 and Figure 14 show that TDL often yields remarkable improvements. Specifically, we observe that TDL's impact seems to depend on two factors: (1) in how far it is able to resolve issues of the baseline (e.g., the dimensional collapse of GraphCL), and (2) in how far the PIs used for TDL "suit" the dataset considered (see also Figure 8). We observe that the latter yields a task which needs to be solved together with domain experts (i.e., finding the best knowledge to use to create PIs).
F Unsupervised Learning
Datasets. In the unsupervised context, we adopted various molecular benchmarks from TUDataset [Morris et al., 2020], including NCI1, MUTAG, PROTEINS, and DD. Table 15 shows statistics for datasets.
Table 15: Summary for molecular datasets from the benchmark TUDataset.
Dataset	# Molecules	# Class	Avg. # Nodes	Avg. # Edges
NCI1	4,110	2	29.87	32.30
MUTAG	188	2	17.93	19.79
PROTEINS	1,113	2	39.06	72.82
DD	1,178	2	284.32	715.66
Configurations. For the unsupervised graph classification task, we first train a representation model contrastively using unlabeled data, then fix the representation model and train the classifier using labeled data. Following GraphCL [You et al., 2020], we employ a 5-layer GIN with a hidden size of 128 as our representation model and utilize an SVM as our classifier. The GIN is trained with a batch size of 128 and a learning rate of 0.001. Regarding graph representation learning, models are trained for 20 epochs and tested every 10 epochs. We conduct a 10-fold cross-validation on every dataset. For each fold, we utilize  $90\%$  of the total data as the unlabeled data for contrastive pre-training and the remaining  $10\%$  as the labeled testing data. Every experiment is repeated 5 times using different random seeds, with mean and standard deviation of accuracies  $(\%)$  reported. We apply TDL on GraphCL and SimGRACE. The temperature parameter  $\tau$  for TDL here is chosen from  $\{0.1, 0.2, 0.5, 1.0, 10.0\}$ . Besides adding a TDL to the loss function in CL approaches, we did not make any modifications to the models.
Results. Table 16 displays the comparison among different models for unsupervised learning, revealing that a general performance improvement is achieved with the TDL. We also compute the normalized Euclidean distance of embeddings from these unsupervised datasets, and the histogram is illustrated in Figure 15. The findings are consistent with our prior observations in transfer learning; however, the impact may not be as substantial due to the smaller number of training samples.
Table 16: Results of unsupervised learning. For each datasets, we report the mean (and standard deviation) accuracies of 5 seeds.
NCI1	MUTAG	DD	PROTEINS
GraphCL	77.87 (0.41)	86.80 (1.34)	78.62 (0.40)	74.39 (0.45)
GraphCL + TDLatom	80.06 (0.37)	89.12 (0.79)	79.88 (0.47)	75.59 (0.48)
SimGRACE	79.12 (0.44)	89.01 (1.31)	77.44 (1.11)	75.35 (0.09)
SimGRACE + TDLatom	80.08 (0.31)	89.47 (1.09)	79.01 (0.84)	75.39 (0.57)
G Ablation Studies
Different Filtrations - Geometry. Firstly, we consider detailed ablations of the filtration functions used in the graph filtration step of  $\mathrm{TAE_{ahd}}$ . That is, we use individual PIs (i.e., each PI is/was constructed based on a separate filtration function) instead of concatenation as reconstruction targets for  $\mathrm{TAE_{ahd}}$ . Consequently, we obtain 3 individual TAEs based on different PIs. We also conduct an experiment that involves integrating these 3 TAEs during the fine-tuning of downstream tasks. More specifically, we individually fine-tune these models, concatenate their embeddings after the readout, and finally project through a projection head for downstream task prediction. As demonstrated in Table 17, concatenation during fine-tuning (FT) does indeed combine the advantages of individual filtrations. These filtrations are very generic and the method is flexible in terms of what is useful downstream. That is, this approach would allow domain experts to flexibly use their own filtrations in combinations and together with existing methods, in dependence on the specific downstream data. This also suits the well-known fact that specific downstream data may benefit from specific domain knowledge.
Figure 15: Evaluating alignment in unsupervised datasets. Dark/light shades denote positive/negative pairs, respectively. Blue illustrates the embedding of baselines, red depicts the embedding of baselines  $+\mathrm{TDL}_{\mathrm{atom}}$ , and green represents  $\mathrm{PI}_{\mathrm{atom}}$ .
For TDL, the filtration based on atom number provided overall best performance; we ran all experiments on PIs concatenated as above as well. We plan to investigate more in this direction in the future.
Different Filtrations - Domain Knowledge. Furthermore, we show results about incorporating different kinds of domain knowledge, here, atomic radius and electron affinity, as filtration functions. For TAE, we conduct two experiments: (1) We pre-train TAE using individual PIs generated by atomic radius and electron affinity as reconstruction targets; (2) We concatenate these 5 PIs (with the previous 3 PIs) as reconstruction targets. For TDL, we use these two individual PIs as topological fingerprints and apply TDL on GraphCL. Table 17 shows that these two domain knowledge-based filtration functions achieve commendable average performance. Yet, in summary, we do not observe considerable differences. Our hypothesis is that, similar to the atomic number, these kinds of domain knowledge capture the difference between the atoms, and hence are not too different overall. Based on this investigation, we plan to experiment with more complex sources of domain knowledge (e.g., the polarity of atoms, which is important information for the possible relationships between atoms).
Different Filtrations - Multiple Dimensions. Here we consider the multidimensional Vietoris-Rips (multi-VR) filtration [Demir et al., 2022]. And we use the atomic number, heat kernel signature with  $t = 0.1$  and node degree as filtration functions in the first dimension, respectively. Then we compute PDs and then vectorize them to get 3 PIs. Finally, we concatenate these 3 PIs as reconstruction targets to pre-train TAE.
No Filtrations. Please observe first that our study's (initial) research question has been the exploration of the usefulness of PH and topological fingerprints for molecular SSL. Based on our investigation (see also Figure 8), there is one very interesting outcome of our research and follow-up research question related to TDL: TDL is an instance of a more general proposal "XDL", a novel loss function for pre-training which is exploits distances between training samples. We conducted some initial experiments only using the Tanimoto distance between standard ECFP fingerprints and obtain decent results; see Table 17. In future research, we will investigate the generality and potential impact of XDL more (e.g., combining ECFPs and topological fingerprints).
Different Vectorizations of PDs. Persistence Vectorizations transform the obtained Persistent Homology information (PDs) into either a functional or a feature vector form. This representation is more suitable for ML tools compared to PDs. In our study, we employ Persistence Images for vectorization. Apart from this, we also explored other stable vectorizations for TAE, such as Persistence Landscapes [Bubenik et al., 2015] and Silhouettes [Chazal et al., 2014]. As shown in Table 18, ablation experiments demonstrate that Persistence Images outperform the other two vectorization techniques.
Hyperparameters Analysis. We further perform a sensitivity analysis on the parameter  $\lambda$  of GraphCL + TDL $_{\text{ToDD}}$  across 8 downstream molecular tasks. The results are presented in Table 19. It
is noteworthy that as  $\lambda$  decreases, the weight of TDL also decreases, leading to a noticeable decline in performance. This observation underscores the effectiveness of TDL.
Table 17: Ablation study results for 8 molecular property prediction tasks on ZINC15. For each downstream task, we report the mean (and standard deviation) ROC-AUC of 10 seeds.
Tox21	ToxCast	Sider	ClinTox	MUV	HIV	BBBP	Bace	Average
No pretrain	74.6 (0.4)	61.7 (0.5)	58.2 (1.7)	58.4 (6.4)	70.7 (1.8)	75.5 (0.8)	65.7 (3.3)	72.4 (3.8)	67.15
TAEahd	75.2 (0.8)	63.1 (0.3)	61.9 (0.8)	80.6 (1.9)	74.6 (1.8)	73.5 (2.1)	67.5 (1.1)	82.5 (1.1)	72.36
TAEatom	74.6 (0.3)	61.8 (0.6)	62.5 (0.7)	71.0 (3.8)	75.7 (2.1)	75.5 (0.8)	67.0 (1.1)	78.6 (1.3)	70.84
TAEhks	75.5 (0.3)	62.7 (0.3)	59.0 (0.6)	79.3 (3.0)	75.5 (1.4)	73.5 (1.2)	68.6 (0.5)	82.5 (0.8)	72.08
TAEdegree	75.6 (0.6)	62.8 (0.5)	58.2 (0.8)	80.0 (1.3)	70.7 (2.0)	73.4 (1.6)	68.1 (0.8)	81.9 (0.6)	71.34
TAEConcatenation during FT	75.7 (0.6)	64.2 (0.4)	61.9 (1.2)	80.2 (2.1)	74.4 (2.9)	75.1 (1.1)	68.1 (0.8)	82.1 (1.1)	72.71
TAEAtomic radius	76.5 (0.6)	63.3 (0.7)	63.2 (1.4)	76.2 (2.6)	71.9 (2.7)	74.3 (1.3)	67.4 (1.3)	80.9 (1.7)	71.71
TAEElectron affinity	75.6 (0.6)	63.0 (0.7)	64.0 (0.7)	78.5 (1.9)	72.6 (2.8)	76.3 (1.6)	68.9 (1.0)	81.5 (1.3)	72.55
TAEConcatenation of 5 PIs	75.9 (0.4)	63.8 (0.3)	62.5 (0.6)	81.6 (2.4)	75.6 (1.4)	74.2 (0.9)	66.6 (0.9)	82.7 (1.2)	72.86
TAEMulti-VR filtration	75.8 (0.6)	63.4 (0.6)	62.0 (0.7)	76.8 (2.6)	75.1 (1.2)	76.4 (0.6)	69.9 (0.9)	81.7 (1.3)	72.64
TAEToDD	76.8 (0.9)	64.0 (0.5)	61.9 (0.8)	79.3 (3.6)	75.8 (3.2)	75.9 (1.1)	70.4 (0.8)	81.6 (1.4)	73.22
GraphCL	75.0 (0.3)	62.8 (0.2)	60.1 (1.3)	78.9 (4.2)	77.1 (1.0)	75.0 (0.4)	67.5 (3.3)	68.7 (7.8)	70.64
GraphCL + TDLatom	75.3 (0.4)	64.4 (0.3)	61.2 (0.6)	83.7 (2.7)	75.7 (0.8)	78.0 (0.9)	70.9 (0.6)	80.5 (0.8)	73.71
GraphCL + TDLAtomic radius	74.3 (0.3)	63.2 (0.3)	61.3 (0.4)	85.5 (1.5)	77.6 (2.4)	77.3 (1.3)	70.7 (0.6)	74.3 (1.3)	73.02
GraphCL + TDLElectron affinity	75.1 (0.3)	63.3 (0.4)	60.2 (0.6)	84.9 (3.0)	76.0 (1.4)	78.8 (1.9)	70.1 (0.6)	79.5 (1.1)	73.50
GraphCL + TDLToDD	75.2 (0.7)	64.2 (0.3)	61.5 (0.4)	85.2 (1.8)	75.9 (2.1)	77.9 (0.8)	69.9 (0.9)	81.2 (1.9)	73.88
GraphCL + TDLECFP	75.3 (0.4)	64.0 (0.2)	61.1 (0.6)	82.6 (2.8)	75.0 (1.1)	76.4 (0.9)	70.1 (0.7)	78.3 (1.7)	72.85
Table 18: Comparison of the different vectorizations.
Tox21	ToxCast	Sider	ClinTox	MUV	HIV	BBBP	Bace	Average
TAEadh (Persistence Images)	75.2 (0.8)	63.1 (0.3)	61.9 (0.8)	80.6 (1.9)	74.6 (1.8)	73.5 (2.1)	67.5 (1.1)	82.5 (1.1)	72.36
TAEadh (Persistence Landscapes)	75.6 (0.5)	63.9 (0.5)	59.5 (0.8)	73.5 (2.9)	73.6 (0.9)	74.0 (1.4)	69.7 (1.6)	79.8 (1.4)	71.21
TAEadh (Silhouettes)	74.7 (0.6)	63.4 (0.6)	58.4 (0.7)	74.1 (2.4)	72.2 (1.6)	75.8 (1.4)	68.9 (0.9)	79.5 (1.4)	70.88
Table 19: Sensitivity analysis on the parameter  $\lambda$  .
GraphCL + TDLToDD	Tox21	ToxCast	Sider	ClinTox	MUV	HIV	BBBP	Bace	Average
λ = 0.1	74.2 (0.8)	63.1 (0.2)	60.8 (0.6)	76.4 (5.8)	73.3 (1.7)	77.2 (1.6)	70.2 (0.2)	71.6 (0.9)	70.85
λ = 0.5	74.0 (0.5)	63.2 (0.3)	60.0 (0.4)	78.6 (0.1)	73.6 (1.4)	77.2 (0.6)	69.9 (0.3)	76.2 (1.7)	71.59
λ = 1.0	75.2 (0.7)	64.2 (0.3)	61.5 (0.4)	85.2 (1.8)	75.9 (2.1)	77.9 (0.8)	69.9 (0.9)	81.2 (1.9)	73.88
λ = 2.0	74.8 (0.3)	63.6 (0.3)	61.7 (0.7)	85.8 (1.7)	74.9 (0.8)	77.2 (1.0)	71.0 (0.7)	81.8 (1.2)	73.85
Different GNN Backbones. We apply TDL on GraphCL. As shown in Table 20, we verify the generality of TDL by experimenting with three popular GNN models: GIN [Xu et al.], GCN [Kipf and Welling], and GraphSAGE [Hamilton et al., 2017]. Pre-training with GIN achieves the best performance across datasets, hence our results confirm what has been observed in related works.
Table 20: Comparison of the pre-training gains, represented in terms of averaged ROC-AUC  $(\%)$ , obtained by using different GNN architectures on 8 datasets.
Model	GCN	GIN	GraphSAGE
No pretrain	68.77	67.15	68.32
GraphCL	69.32	70.78	70.08
GraphCL + TDLatom	72.53	73.71	71.16
H PIs w/o SSL
Here we provide initial results for a comparison with 2 non pre-training ML models, i.e. support vector machine (SVM) and XGB [Chen and Guestrin, 2016] to test the regular prediction on 5 molecular property prediction tasks using PIs as topological fingerprints. And we also conduct smaller data experiments on SVM compared with TAE and TAE+ContextPred over 4 downstream datasets. Note that we performed a hyperparameter search to produce these ML results. As can be observed in Table 21 and Figure 16, the SSL methods provide large benefits. We are certainly aware of the fact that there are more complex methods to use these PIs, yet this basic comparison gives a good hint of how SSL based on PH compares to supervised methods more generally.
Table 21: Results of non pre-training method for molecular property prediction tasks on ZINC15, ROC-AUC (\%).
Tox21	Sider	ClinTox	BBBP	Bace
SVM	53.9	53.8	60.1	56.6	72.3
XGB	64.7	58.5	55.7	61.6	73.8
ContextPred	75.7	60.9	65.9	68.0	79.6
TAEahd + ContextPred	76.4	62.0	74.6	68.9	80.7
Figure 16: Performance of SVM over smaller datas. Each datapoint is the mean ROC-AUC of 10 different splits while keeping scaffolds, with standard deviations shown as error bars. From the left to the right: Bace, BBBP, ClinTox, Sider.
I Theoretical Analysis of TDL
In this section, we delve into a theoretical analysis of TDL, which was initially introduced in Section 3.2. We encourage readers to refer to Section 3.2 for any specific notation, terminology, or formulation details.
Recall TDL is defined as follows:
$$
\mathcal{L}_{\mathrm{TDL}_{n}} = \frac{1}{N - 1}\sum_{\substack{m\in [1,N],\\ m\neq n}} - \log \frac{\exp\left(sim\left(z_{n},z_{m}\right) / \tau\right)}{\sum_{\substack{k\in[1,N],\\ k\neq n}}\mathbb{I}_{[dis(I_{n},I_{k})\geq dis(I_{n},I_{m})]}\cdot\exp\left(sim\left(z_{n},z_{k}\right) / \tau\right)}
$$
For the sake of convenience, we re-index the subscripts of samples, ensuring that the subscripts  $d_{i}$  satisfy the sorted distances from the n-th sample, i.e.,  $dis(I_n,I_n)\leq dis(I_n,I_{d_1})\leq dis(I_n,I_{d2})\leq$ $dis(I_{n},I_{d_{3}})\leq \ldots \leq dis(I_{n},I_{d_{N - 1}})$ . Let  $sim(z_{n},z_{m}) = z_{n}z_{m}$  to streamline the derivations. We thus obtain
$$
\mathcal {L} _ {\mathrm {T D L} _ {n}} = \frac {1}{N - 1} \sum_ {j \in [ 1, N - 1 ]} - \log \frac {\mathrm {e} ^ {(z _ {n} \cdot z _ {d _ {j}} / \tau)}}{\sum_ {k \in [ j , N - 1 ]} \mathrm {e} ^ {(z _ {n} \cdot z _ {d _ {k}} / \tau)}}
$$
We then analyze the gradients with respect to different samples.
Lemma 1. The gradient of  $TDL_{n}$  with respect to the sample  $d_{i}$  in latent space is formulated as:
$$
\nabla_ {z _ {d _ {i}}} \mathcal {L} _ {T D L _ {n}} = \frac {1}{N - 1} \left(\sum_ {j \in [ 1, i ]} \frac {e ^ {z _ {n} \cdot z _ {d _ {i}} / \tau}}{\sum_ {k \in [ j , N - 1 ]} e ^ {z _ {n} \cdot z _ {d _ {k}} / \tau}} - 1\right) \cdot \frac {z _ {n}}{\tau}.
$$
In particular,  $\nabla_{z_{d_1}}\mathcal{L}_{TDL_n}\propto -z_n$  and  $\nabla_{z_{d_N - 1}}\mathcal{L}_{TDL_n}\propto z_n$
Proof. The results follow from direct computation. Let  $\mathcal{L}_{\mathrm{TDL}_{n,d_j}} = -\log \frac{\mathrm{e}^{(z_n\cdot z_{d_j} / \tau)}}{\sum_{k\in[j,N - 1]}\mathrm{e}^{(z_n\cdot z_{d_k} / \tau)}},$  then we have:
$$
\mathcal {L} _ {\mathrm {T D L} _ {n}} = \frac {1}{N - 1} \sum_ {j \in [ 1, N - 1 ]} - \log \frac {\mathrm {e} ^ {\left(z _ {n} \cdot z _ {d _ {j}} / \tau\right)}}{\sum_ {k \in [ j , N - 1 ]} \mathrm {e} ^ {\left(z _ {n} \cdot z _ {d _ {k}} / \tau\right)}} = \frac {1}{N - 1} \sum_ {j \in [ 1, N - 1 ]} \mathcal {L} _ {\mathrm {T D L} _ {n, d _ {j}}}
$$
Next, we discuss  $\nabla_{z_{d_i}}\mathcal{L}_{\mathrm{TDL}_{n,d_j}}$
$$
\nabla_ {z _ {d _ {i}}} \mathcal {L} _ {\mathrm {T D L} _ {n, d _ {j}}} = \left\{ \begin{array}{l l} 0 & i <   j \\ \left(\frac {e ^ {z _ {n} \cdot z _ {d _ {i}} / \tau}}{\sum_ {k \in [ j , N - 1 ]} \mathrm {e} ^ {z _ {n} \cdot z _ {d _ {k}} / \tau}} - 1\right) \cdot \frac {z _ {n}}{\tau} & i = j \\ \left(\frac {e ^ {z _ {n} \cdot z _ {d _ {i}} / \tau}}{\sum_ {k \in [ j , N - 1 ]} \mathrm {e} ^ {z _ {n} \cdot z _ {d _ {k}} / \tau}}\right) \cdot \frac {z _ {n}}{\tau} & i > j \end{array} \right.
$$
Therefore, we conclude:
$$
\nabla_ {z _ {d _ {i}}} \mathcal {L} _ {\mathrm {T D L} _ {n}} = \frac {1}{N - 1} \nabla_ {z _ {d _ {i}}} \sum_ {j \in [ 1, N - 1 ]} \mathcal {L} _ {\mathrm {T D L} _ {n, d _ {j}}} = \frac {1}{N - 1} \left(\sum_ {j \in [ 1, i ]} \frac {e ^ {z _ {n} \cdot z _ {d _ {i}} / \tau}}{\sum_ {k \in [ j , N - 1 ]} e ^ {z _ {n} \cdot z _ {d _ {k}} / \tau}} - 1\right) \cdot \frac {z _ {n}}{\tau}
$$
$$
\nabla_ {z _ {d _ {1}}} \mathcal {L} _ {\mathrm {T D L} _ {n}} = \frac {1}{N - 1} \left(\frac {e ^ {z _ {n} \cdot z _ {d _ {1}} / \tau}}{\sum_ {k \in [ 1 , N - 1 ]} \mathrm {e} ^ {z _ {n} \cdot z _ {d _ {k}} / \tau}} - 1\right) \cdot \frac {z _ {n}}{\tau}
$$
The quantity  $\left(\frac{e^{z_n\cdot z_{d_1} / \tau}}{\sum_{k\in[1,N - 1]}\mathrm{e}^{z_n\cdot z_{d_k} / \tau}} - 1\right) < 0$  is a strictly negative scalar, allowing us to conclude the derivative  $\nabla_{z_{d_1}}\mathcal{L}_{\mathrm{TDL}_n}$  is proportional to  $-z_{n}$ .
$$
\nabla_ {z _ {d _ {N - 1}}} \mathcal {L} _ {\mathrm {T D L} _ {n}} = \frac {1}{N - 1} \left(\sum_ {j \in [ 1, N - 2 ]} \frac {\mathrm {e} ^ {z _ {n} \cdot z _ {d _ {N - 1}} / \tau}}{\sum_ {k \in [ j , N - 1 ]} \mathrm {e} ^ {z _ {n} \cdot z _ {d _ {k}} / \tau}}\right) \cdot \frac {z _ {n}}{\tau}
$$
Since  $\left(\sum_{j\in [1,N - 2]}\frac{e^{z_n\cdot z_{d_N - 1} / \tau}}{\sum_{k\in[j,N - 1]}\mathrm{e}^{z_n\cdot z_{d_k} / \tau}}\right) > 0$  we conclude in this case that the derivative  $\nabla_{z_{d_{N - 1}}}\mathcal{L}_{\mathrm{TDL}_n}$  points in the direction  $z_{n}$ .
The stability provided by certain topological fingerprints serves as a mathematically grounded, well-studied, and efficient proxy for stability with respect to graphs. TDL as our objective is based on the stability of these topological fingerprints. Consequently, for each sample, we can determine a ranking of all other samples. Lemma 1 provides the gradient with respect to different samples. Specifically, samples that are distant from sample  $n$  in the topological space point in the direction  $-z_{n}$  (negative gradient direction), while those close to sample  $n$  point in the direction  $z_{n}$ . Consequently, molecules with more similar topological features are brought closer to each other in the embedding space.
J Potential of PH for Molecular ML
Persistent homology (PH) has emerged as a crucial method in topological data analysis, demonstrating significant potential in decoding complex patterns within molecular data, including protein prediction [Cang and Wei, 2017a,b, Nguyen et al., 2019, Xia and Wei, 2014, Liu et al., 2022b], virtual screening [Cang et al., 2018], and drug design [Liu et al., 2021]. The many works on PH in chemistry show that the domain is convinced by the usefulness of topological fingerprints already, and we therefore believe that this kind of knowledge should be investigated in the context of SSL, assuming that the latter becomes more important with the advancement of foundation models.
K Limitations & Broader Impact
Broader Impact. Graph data has a pervasive impact in many diverse fields and the popularity certainly increases the danger of an application of the models with negative effects (even if just due to ignorance). As pointed out in this work w.r.t. smaller data, we have to be aware of the assumptions we make in our research to inform readers on how the results translate to the real world. Our goal is to advance biomedicine, chemistry, and material science, where our research hopefully provides benefits one day, for both the current society and future generations.
Limitations. We have tried our best in addressing common issues, by providing a very detailed analysis, by considering TDL on top of various baselines, by also reporting mixed results, and by providing all code. However, due to time and resource constraints, and the broadness of our study, we were not able to experiment with a variety of topological fingerprints, analyze individual baselines (e.g., the dimensional collapse) in more detail, or complete the theoretical investigations.