SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Figure 38: Histograms of sample means for different features
Figure 39: Histograms of sample variance for different features
Figure 40: Histograms of energy weighted averages
Figure 41: Histograms of lateral widths
(a) Point dist. for Energy feature
(d) Cell Energy Ratio
Figure 42: Histograms of various shower shape variables
A.5 Experiments on grid representation of data
In this section we will present some studies on generative modeling with the grid representation of data from SUPA. We discuss about how to downsample the point clouds below. For these studies, we generated another version of the dataset with SUPA such that it is similar to the CALOGAN dataset, i.e., with three layers and downsampled to a resolution in the multiples of  $3 \times 96$ ,  $12 \times 12$ , and  $12 \times 6$ , for layer 0, 1, and 2, respectively.
Downsampling. For comparison, we downsample the point clouds to their corresponding image representation (see Figure 1) by first defining the region of interest i.e. a rectangular region for each layer and the number of bins/cells/pixels in both the horizontal (or  $\eta$ ) and vertical (or  $\phi$ ) directions. Finally, for each cell, we sum the energy of all the points falling within it to get the pixel intensity. We can increase the number of cells in order to get higher resolutions. Figure 1b, 1c, and 1d show the downsampled image representations at resolutions of  $3x$ ,  $2x$  and  $1x$  respectively for the shower shown in Figure 1a. We choose  $1x$  to be the same resolution as used in CaloGAN [Paganini et al., 2018] (i.e.  $12 \times 12$  for Layer 1).
A.5.1 Validity of SUPA as a benchmark with grid representation
We show the comparison of performance of generative models trained over data generated with SUPA and Geant4 in § 5.3. In this section, we extend those studies with more analysis and plots. Figure 43 shows the scatter plot of the average ranks of those models. The average rank for a model on a dataset is obtained by first ranking them with respect to each marginal's discrepancy and then averaging over all the marginals.
Further, in Figures 44-49, we show a subset of the marginals (see § 5 for a detailed explanation on the marginals and Paganini et al. [2018] for the grid representation based marginals) for GEANT4 and
Figure 43: Scatter Plot for ranks over different models. Ranking of the models are consistent over both, SUPA and GEANT4, showing the validity of SUPA as a benchmark.
Figure 44: Histogram for various margins for GEANT4 e+ (top) and SUPA (bottom) vs. showers generated from different trained models
SUPA and also the showers generated with different models trained on them. These marginal plots illustrate the diversity in various distributions present in data from GEANT4, and, more importantly in SUPA. Further, the distributions of the generated showers from different models behave similarly on both datasets, reinstating the proposition that a better model on SUPA implies a better model on the detailed GEANT4.
Figure 45: Histogram for Layer Energy for GEANT4 e+ (top) and SUPA (bottom) vs. showers generated with different trained models.
Figure 46: Histogram for Layer energy fraction for GEANT4 e+ (top) and SUPA (bottom) vs. showers generated with different trained models.
Figure 47: Histogram for Layer lateral width for GEANT4 e+ (top) and SUPA (bottom) vs. showers generated with different trained models.
A.5.2 High-resolution experiments
In this section, we show the utility of SUPA beyond using it for training at low resolution (similar to the resolution used in CaloGAN, which we call 1x), as well as the limitation of the current models.
1x	2x	3x
1x	3.57	6.35	7.20
2x	-	6.78	-
3x	-	-	8.29
Table 4: Mean discrepancy metric (see § A.5.1) for CaloFlow model when trained and tested over different resolutions. Columns correspond to the training resolution and rows to the test resolution. The results on the diagonal show that CaloFlow's performance degrades when resolution increases, and the top row shows that it is not simply due to the sheer dimensionality of the signal since the model does not leverage structure at high resolution to perform better at low resolution.
We train CaloFlow [Krause and Shih, 2021] with SUPA by downsampling the point clouds at the higher resolutions of  $2\mathrm{x}$  and  $3\mathrm{x}$ . Table 4 shows the mean discrepancy metric (see  $\S$  A.5.1) for the models. We observe the trend that training at higher resolutions result in poorer performance (diagonal terms) in general. Further, when the generated samples from the trained models are downsampled to  $1\mathrm{x}$ , the performance deteriorates as compared to samples generated from models trained directly with data at  $1\mathrm{x}$  resolution.
Figure 48: Histogram for  $E_{\mathrm{ratio},i}$  for GEANT4 e+ (top) and SUPA (bottom) vs. showers generated with different trained models.
Figure 49: Histogram for Layer sparsity for GEANT4 e+ (top) and SUPA (bottom) vs. showers generated with different trained models.