SlowGuess/NeurIPS_2023_OCR huggingface dataset:
SUPA is very flexible and can be easily extended. The slices can be further divided into different regions each with their own meta-parameters providing more granular control over the scattering and splitting process. Another variation requires having some slices that are not merged into layers to mimic dead material. These extensions can possibly bring it closer to more realistic detectors, and lead to unique structures in the dataset and thus present challenges for model development and evaluation. It is also possible to condition the data generation on various attributes, for example: initial impact angle, impact energy, impact position, etc. This encourages interesting studies and benchmarks regarding extrapolation and generalization capabilities of generative models for unseen conditioning variables during training.
5 Evaluation and Analysis
The evaluation of generative models for high dimensional data is a non-trivial problem. While reconstruction losses and/or data likelihood are available for some model architectures (e.g. autoencoders or flows), judging the sample quality is still difficult. Moreover, the average log-likelihood is difficult to evaluate or even approximate for many interesting models. For instance, in computer vision, evaluating generative models for images is done via proxy measures such as the Frechet Inception Distances (FID, Heusel et al., 2017). Point cloud generative models are evaluated [Yang et al., 2019] using metrics such as Minimum matching distance (MMD), Coverage (COV) and 1-Nearest Neighbour Accuracy (1-NNA), where similarity between a set of reference point clouds and generated point clouds are measured using a distance metric such as the Chamfer distance (CD) or Earth mover's distance (EMD) based on optimal matching.
5.1 Shower Shape Variables
For calorimeter simulation, we cannot judge performance using individual examples due to the stochastic nature of shower development. The standard practice to estimate the quality of generative models for calorimetry data (shower simulations) is to use histograms of different metrics called shower shape variables [Paganini et al., 2018] in addition to images of calorimeter showers. These shower shape variables are domain-specific and physically motivated, and matching densities over these marginals is indicative of matching salient aspects of the data. These marginals capture various aspects of how energy is distributed within individual layers and across different layers.
We extend the shower shape variables for the point clouds representation of showers. A summary of all the shower shape variables considered are present in the appendix.
Point level marginals. Marginals of each point feature by considering the set all the points from all the point clouds together.
Feature means  $\langle \eta_i\rangle ,\langle \phi_i\rangle ,\langle r_i\rangle ,\langle E_i\rangle$  . Mean of each feature.
$$
\langle \eta_ {i} \rangle = \frac {\sum_ {j} \eta_ {j} ^ {i}}{\sum_ {j} 1}, \langle \phi_ {i} \rangle = \frac {\sum_ {j} \phi_ {j} ^ {i}}{\sum_ {j} 1}, \langle r _ {i} \rangle = \frac {\sum_ {j} r _ {j} ^ {i}}{\sum_ {j} 1} \langle E _ {i} \rangle = \frac {\sum_ {j} E _ {j} ^ {i}}{\sum_ {j} 1}
$$
where  $r_j^i = \sqrt{(\eta_j^i)^2 + (\phi_j^i)^2}$  denotes the distance of the point in the lateral plane from the center.
Feature variances  $\sigma_{\langle \eta_i\rangle},\sigma_{\langle \phi_i\rangle},\sigma_{\langle r_i\rangle},\sigma_{\langle E_i\rangle}$ . Variance of each feature.  $\sigma_{\langle \eta_i\rangle} = \sqrt{\frac{\sum_j\eta_j^{i^2}}{\sum_j1} - \langle\eta_i\rangle^2}$
Layer Energy  $\bar{E}_i$ . Denotes the total energy deposited in layer  $i$  of the shower.  $\bar{E}_i = \sum_{j \in N_i} E_j^i$ .
Total Energy  $E_{\mathrm{tot}}$ . Total energy across all layers of the shower.  $E_{\mathrm{tot}} = \sum_{i \leq N} \bar{E}_i$
Layer Centroids  $\langle \eta_i\rangle_E,\langle \phi_i\rangle_E,\langle r_i\rangle_E$ . Energy weighted mean of the features  $(\eta ,\phi ,$  or  $r)$
$$
\langle \eta_ {i} \rangle = \frac {\sum_ {j} E _ {j} ^ {i} \eta_ {j} ^ {i}}{E _ {i}}, \langle \phi_ {i} \rangle = \frac {\sum_ {j} E _ {j} ^ {i} \phi_ {j} ^ {i}}{E _ {i}}, \langle r _ {i} \rangle = \frac {\sum_ {j} E _ {j} ^ {i} r _ {j} ^ {i}}{E _ {i}}
$$
The layer centroids can be interpreted as the center of energy in the lateral plane in respective dimensions.
Layer Lateral Width  $\sigma_{\langle \eta_i\rangle_E},\sigma_{\langle \phi_i\rangle_E},\sigma_{\langle r_i\rangle_E}$ . Denotes the standard deviation of the layer centroids.
$$
\sigma_ {\langle \eta_ {i} \rangle_ {E}} = \sqrt {\frac {\sum_ {j} E _ {j} ^ {i} (\eta_ {j} ^ {i}) ^ {2}}{E _ {i}} - \langle \eta_ {i} \rangle_ {E} ^ {2}}
$$
The layer lateral widths can be interpreted as the spread around the center of energy in the lateral plane in respective dimensions. We drop the layer notation  $i$  from the above metrics when working with a single layer for brevity.
5.2 Performance Analysis of Generative Models
In order to summarise the discrepancy into a scalar value, we consider the Wasserstein-1 distance between the histograms of the ground truth and generated sample's marginals<sup>5</sup>. Further, we compute the mean discrepancy over different groups of marginals to summarize the performance of models.
We train point cloud generative models, PointFlow [Yang et al., 2019], SetVAE [Kim et al., 2021], and a transformer-based flow model (which we call Transflowmer, see Sec. A.3), on SUPA data. Table 3 summarizes the performance of different models across different variations of SUPA datasets. Pointflow performs better than SetVAE for all marginals not involving energy feature such as point level marginals, feature means and variances for location parameters, etc., and worse for marginals which depend on energy such as, layer centroids, and layer energy, etc. (with the exception of layer lateral widths and feature variance  $\sigma_{\langle E\rangle}$ , which are both second order moments), across all datasets. Roughly speaking, PointFlow struggles at modeling the energy feature while it is competitive for the location features, and SetVAE behaves opposite. It is interesting to observe that Transflowmer is either the best performing or very close to best performing model (either SetVAE or PointFlow) across all datasets and marginals, indicating relatively greater flexibility at modeling a variety of features. The instances with relatively high values in Table 3 (feature means for SUPAv1, and layer energy for SUPAv2 and SUPAv3) are where the marginals had a very narrow spread, indicating the difficulty in modeling discrete distributions. The models also have the tendency to predict points with very high energy values unseen during training (see Fig. 24a). We provide more details in the Appendix Section A.4 on the training, results and analysis, along with plots of various shower shape variables to visually compare the goodness of fit.
These observations highlight the need for adapting point cloud generative models such that they are able to model points with special features (different from location parameters), such as physical features like energy, momentum, etc. Although the location parameters are correlated with these special features, the overall structure of this correlation can be very different from domain to domain and also from how they are correlated with other location features.
For computing the Wasserstein distance, we normalize the shower shape variables for the ground truth and the generated sample according to the mean and standard deviation of the ground truth.
Table 3: Performance benchmarks across different datasets with SetVAE, PointFlow and Transformer. The distance metric is Wasserstein-1. The reported numbers are averages over a group of marginals as indicated in the top row. Lower numbers are better. See § 5.1 for more details on the metrics.
Point Features : η, φ, r	Point Feature : E	<ηi>, <φi>, <ri>
Dataset	SV	PF	TF	SV	PF	TF	SV	PF	TF
SUPAv1	0.044	0.037	0.028	0.001	0.000	0.000	16.254	21.921	5.789
SUPAv2	0.290	0.167	0.029	0.253	0.462	0.145	0.517	0.256	0.091
SUPAv3	0.510	0.130	0.044	0.304	0.484	0.048	0.769	0.075	0.105
SUPAv4	0.269	0.122	0.029	0.111	0.459	0.006	0.460	0.094	0.073
SUPAv5	0.166	0.028	0.042	0.078	0.090	0.151	0.592	0.031	0.189
σ(ηi), σ(φi), σ(ri)	< E>	σ(E)
Dataset	SV	PF	TF	SV	PF	TF	SV	PF	TF
SUPAv1	0.513	0.585	0.359	0.001	0.000	0.000	0.000	0.000	0.000
SUPAv2	0.648	0.154	0.130	0.302	0.077	0.087	0.513	0.177	0.165
SUPAv3	1.114	0.109	0.126	0.320	0.064	0.071	0.500	0.152	0.144
SUPAv4	0.634	0.092	0.051	0.263	0.047	0.022	0.355	0.156	0.038
SUPAv5	0.799	0.040	0.223	0.251	0.059	0.414	0.377	0.047	0.421
⟨ηi⟩E, ⟨φi⟩E, ⟨ri⟩E	σ⟨ηi⟩E, σ⟨φi⟩E, σ⟨ri⟩E	E̅
Dataset	SV	PF	TF	SV	PF	TF	SV	PF	TF
SUPAv1	16.244	21.921	5.766	0.517	0.591	0.379	0.101	0.000	0.039
SUPAv2	1.336	1.933	0.137	0.779	0.134	0.190	23.387	69.814	10.468
SUPAv3	1.365	1.627	0.217	1.226	0.147	0.151	36.116	79.062	5.916
SUPAv4	1.369	1.346	0.083	0.645	0.169	0.067	3.997	12.895	0.659
SUPAv5	2.373	1.615	0.463	0.740	0.058	0.317	6.132	16.742	9.273
5.3 SUPA as a benchmark
We train the models of § 2.2 on both the events generated by GEANT4 and SUPA. For these studies, we generated another version of the dataset with SUPA and downsample the point clouds to grid representation such that it is similar to the CALOGAN dataset (see § A.5 for details on data generation). The training loss of the flow model is a log-likelihood which meaningfully represents the quality of the captured distribution. Figure 3 displays the correlation of the log-likelihood values of different variations of the CaloFlow architecture trained both on GEANT4 and SUPA-generated events. Figure 4 shows the scatter plot of the mean discrepancy (across all margins) obtained by different models on GEANT4 and SUPA-generated events. We show more plots in the Appendix § A.5.
Figure 3: Scatter plot for loss over different versions of CaloFlow models. Each point corresponds to an architecture, its  $x$  and  $y$  coordinates are the negative log-likelihoods on data generated by GEANT4 and SUPA, respectively. Point size reflects the capacity of the model (number of layers, learnable parameters, etc.).
Figure 4: Scatter Plot for mean discrepancy (\$5) over different models. Lower numbers are better. Performance of models are consistent over both datasets, a better model on SUPA implies a better model on GEANT4 and vice versa.
We can observe that all these correlation plots are roughly monotonic, showing that benchmarking on SUPA provides a good estimate of relative performance on the detailed GEANT4.
We introduced SUPA, a lightweight pseudo-particle simulator that is inspired by the physics governed development of particle showers and qualitatively resembles the data generated by the gold standard GEANT4. By allowing to easily change the underlying parameters of the propagation model, we introduce the freedom to define new benchmark datasets of varying complexity with a fidelity and simplicity not available using the GEANT4 toolkit.
We believe that the ease of use, flexibility and speed of this simulator could be a MNIST moment for the field, allowing the rapid exploration of models in various regime of functioning as discussed in § 4.1. We also highlight the gaps in the ability of existing point cloud generative models on calorimetry data in § 5.2 and highlight the need for more powerful point cloud generative models.
Additionally, we showed in § 5.3 and § A.5.1 that with grid representation of the shower data, performance of deep generative models estimated on data produced with this simulator is a good proxy for their performance on the standard data used historically in machine learning for particle physics. We highlight flexibility of SUPA in § A.5.2, allowing us to train/evaluate at different resolutions.
Moreover, by giving easy access to the fully recorded event, we open the way to applying methods from point clouds and geometric deep learning research, with the goal of possibly devising end-to-end architectures able to reconstruct a full event from the energy deposits by solving the complete inverse problem, something currently impossible with data from GEANT4.
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A.1 Shower shape variables
We extend the list of shower shape variables described in Sec. 5.1 :
Layer Energy Fraction  $f_{i}$ . Fraction of the total energy deposited in layer  $i$  of the shower.  $f_{i} = \overline{E_{i}} / E_{\mathrm{tot}}$ .
Energy Ratio  $E_{\text{ratio},i}$ . Ratio of the difference between highest and second highest energy intensity point or cell in layer  $i$  and their difference.  $E_{\text{ratio},i} = \frac{E_{[1]}^i - E_{[2]}^i}{E_{[1]}^i + E_{[2]}^i}$ .
Depth  $d$ . Deepest layer in the shower with non-zero energy deposit.  $d = \max_{i}\{i:\max_{j}(E_{j}^{i}) > 0\}$
Layer/Depth Weighted Total Energy  $l_{d}$ . Sum of the layer energies weighted by the layer number.  $l_{d} = \sum_{i\leq N}i\cdot \bar{E}_{i}$
**Shower Depth  $s_d$ .** Depth weighted total energy normalized by the total energy in the shower.  $s_d = l_d / E_{\mathrm{tot}}$
Shower Depth Width  $\sigma_{s_d}$ . Standard deviation of  $s_d$  in units of layer number.
$$
\sigma_ {s _ {d}} = \sqrt {\frac {\sum_ {i = 0} ^ {2} i ^ {2} \cdot \bar {E} _ {i}}{E _ {\mathrm {t o t}}} - \left(\frac {\sum_ {i = 0} ^ {2} i \cdot \bar {E} _ {i}}{E _ {\mathrm {t o t}}}\right) ^ {2}}
$$
Brightest Voxels  $\mathrm{E}_{k}$  brightest_layer. The  $k^{\mathrm{th}}$  brightest voxel in layer  $i$  normalized by the total layer energy.  $\mathrm{E}_{k}$  brightest_layer  $i = E_{[k]}^{i} / E_{i}$ .
Layer Sparsity. The ratio of the number of cells with non-zero energy to the total number of cells in layer  $i$ . This is only valid for the image representation of showers.
A.2 Details on different variations of SUPA datasets
A.2.1 Parameters
Fig. 5 shows the remaining parameters used for generating SUPA variations (see Table 2 for details on other parameters). SUPAv1 is most deterministic as particles always split in the first six sub-layers with no deposits ( $p_{split} = 1$  and  $p_{stop} = 0$  for all sub-layers  $< 7$ ), further since  $p_{stop} = 1$  at sub-layer 7, all the particles get deposited. Thus each event/example in SUPAv1 has exactly  $128(= 2^7)$  points. Further, since  $\alpha$  is fixed to 0, all splits are symmetric and energy is always halved at each split, thus all deposits have the same energy value. SUPAv5 has higher  $p_{split}$  in the initial sub-layers  $(< 7)$  than SUPAv2-4, while  $p_{stop}$  is the same for all of them, thus SUPAv5 has more number of hits/points than SUPAv2-4 in the respective sub-layers or layers.
(a) SUPAv1
(b) SUPAv2, SUPAv3, SUPAv4
(c) SUPAv5
Figure 5: Parameters  $p_{split}$ ,  $p_{stop}$ ,  $p_{pass}$  for SUPA variations
A.2.2 Shower Shape Variables
Fig. 6 shows the average events for different variations of SUPA datasets and Figs. Fig. 7 - 12 shows the histograms of the various shower shape variables for all SUPA datasets.
Figure 6: Average event representation for different variations of SUPA datasets
Figure 7: Histograms of point level distributions
Figure 8: Histograms of feature means
Figure 9: Histograms of feature variances
(a) Point dist. for Energy feature
(d) Energy Variance
(b) Number of Points
(e) Cell Energy Ratio
(c) Energy Mean
(f) Layer Energy
Figure 10: Histograms of various shower shape variables
Figure 11: Histograms of layer centroids
Figure 12: Histograms of layer widths
A.3 Point Cloud Generative Models
PointFlow PointFlow [Yang et al., 2019] is a flow based model with a PointNet-like encoder and a continuous normalizing flow (CNF) decoder. Additionally, the latents (encoder outputs) are modeled with another CNF to enable sampling. We adapted the PointFlow code to handle variable number of points with masking and masked batch norm. The encoder consists of 1D convolutions with filter sizes 128, 128, 256 and 512, followed by a three-layer MLP with 256 and 128 hidden dimensions to convert the point cloud into its latent representation of size 128. The CNF decoder has four conditional concatenations layers with a hidden dimension of 128 and the latent CNF has three concatenations layers with a hidden dimension of 64. The overall architecture has  $0.7M$  trainable parameters.
SetVAE SetVAE Kim et al. [2021] is a transformer-based hierarchical VAE for set-structured data which learns latent variables at multiple scales, capturing coarse-to-fine dependency of the set elements while achieving permutation invariance. We set the number of heads to 4, the dimension of the initial set to 64, the hidden dimension to 64, the number of mixtures for the initial set to 4, and
the number of inducing points in the hierarchical setup to [2, 4, 8, 16, 32]. The overall architecture has  $0.5M$  trainable parameters.
Transflowmer The Transflowmer is flow-architecture using Real NVP layers [Dinh et al., 2016]. As the events are point clouds of varying cardinality, the coupling layers of the flow are required to be permutation equivariant and able to process a varying number of inputs. To satisfy these constraints, we use transformers [Vaswani et al., 2017] without positional encoding in the coupling layers. The overall architecture consists of 16 coupling layers, each of them is parametrised by a 3 transformer layers with  $d_{model} = 32$ . The overall architecture has  $2.1M$  parameters.
We train all the models with  $100K$  training examples.
A.4 Experiments on SUPA datasets
We train point cloud generative models, PointFlow [Yang et al., 2019], SetVAE [Kim et al., 2021], and Transflowmer on SUPA datasets. In this section, we show histogram plots to compare the generative performance across different shower shape variables. For all these plots, the axes limits are chosen according to the ground truth data and generated samples can have probability mass outside the shown range.
A.4.1 SUPAv1
Figs. 13 - 18 show the histograms of various shower shape variables for SUPAv1 and samples generated with PointFlow, SetVAE, and Transflowmer.
Figure 13: Histograms of point distributions for  $\eta, \phi$ , and  $r$
Figure 14: Histograms of sample means for different features
Figure 15: Histograms of sample variance for different features
Figure 16: Histograms of energy weighted averages
Figure 17: Histograms of lateral widths
(a) Point dist. for Energy feature
(d) Cell Energy Ratio
Figure 18: Histograms of various shower shape variables
(b) Energy Mean
(e) Layer Energy
A.4.2 SUPAv2
Figs. 19 - 24 show the histograms of various shower shape variables for SUPAv2 and samples generated with PointFlow, SetVAE, and Transflowmer.
Figure 19: Histograms of point distributions for  $\eta$ ,  $\phi$ , and  $r$
Figure 20: Histograms of sample means for different features
Figure 21: Histograms of sample variance for different features
Figure 22: Histograms of energy weighted averages
Figure 23: Histograms of lateral widths
(a) Point dist. for Energy feature
(d) Cell Energy Ratio
Figure 24: Histograms of various shower shape variables
(b) Energy Mean
(c) Energy Variance
(e) Layer Energy
A.4.3 SUPAv3
Figs. 25 - 30 show the histograms of various shower shape variables for SUPAv3 and samples generated with PointFlow, SetVAE, and Transflowmer.
Figure 25: Histograms of point distributions for  $\eta$ ,  $\phi$ , and  $r$
Figure 26: Histograms of sample means for different features
Figure 27: Histograms of sample variance for different features
Figure 28: Histograms of energy weighted averages
Figure 29: Histograms of lateral widths
(a) Point dist. for Energy feature
(d) Cell Energy Ratio
Figure 30: Histograms of various shower shape variables
A.4.4 SUPAv4
Figs. 31 - 36 show the histograms of various shower shape variables for SUPAv4 and samples generated with PointFlow, SetVAE, and Transflowmer.
Figure 31: Histograms of point distributions for  $\eta, \phi$ , and  $r$
Figure 32: Histograms of sample means for different features
Figure 33: Histograms of sample variance for different features
Figure 34: Histograms of energy weighted averages
Figure 35: Histograms of lateral widths
(a) Point dist. for Energy feature
(d) Cell Energy Ratio
Figure 36: Histograms of various shower shape variables
A.4.5 SUPAv5
We only consider layer 0 for SUPAv5. Figs. 37 - 42 show the histograms of various shower shape variables for SUPAv5 and samples generated with PointFlow, SetVAE, and Transflowmer.
Figure 37: Histograms of point distributions for  $\eta$ ,  $\phi$ , and  $r$