## 3 Formula-Driven Supervised Learning
### FDSL Datasets
#### 3.4.3 Other Notable FDSL Databases  
Another recent FDSL dataset is TileDB [112], which contains patterns made with tiles. A tile is a group of wallpapers with 2D repetition and complicated textures. It is obtained by adding three operations for hexagonally shaped tiles: (i) moving vertices, (ii) deforming edges, and (iii) moving symmetrically in a specific direction. Using this basic technique, the TileDB dataset was created with 1000 classes and 1071 images for each class. The FractalDB, [PERSON] et al. [91] proposed some formula-based datasets for the pretraining of computer-vision models, such as convolutional neural networks [14, 15, 16, 113, 114] and visual transformers [17, 115]. These are the Perlin noise-based _PerlinNoiseDB_ and the Bezier curve-based _BezierCurveDB_. The DeiT model [116] was pretrained and fine-tuned using these formula-based image datasets. This enabled determination that FractalDB was the best choice for FDSL of computer-vision models developed to date. This is confirmed by data presented in Table 2. Improvements in the percentage classification accuracy of up to +18.5, +23.9, +74.4, +21.2 on the Cifar-10, Cifar-100, Cars dataset, and Flowers dataset, respectively, using FractalDB-1k were observed.  
The authors of FractalDB [77] investigated various adaptable parameters, including #category, #instance, filling rate, fractal weight, #dot, and image size. For further information about these parameters, readers may refer to [77]. The experimentation results reported in [77] are shown in Table 3.  
As it can be seen from Table 3, FDSL showed notable performance improvements when using FractalDB for pre-training and its potential was highlighted. From the experimental results presented in Tables 1-3, the performance of FDSL was generally equivalent to that of other competing methods. This performance was achieved using different models pre-trained on synthetic data, in contrast to the huge datasets created by conventional methods.  
\begin{table}
\begin{tabular}{l c c c c} \hline
**Pretraining** & **Learning** & **ModelNet40** & **ModelNet40** & **MIRO** \\
**Technique** & **Method** & **(12 Views)** & **(20 Views)** & **(20 Views)** \\ \hline — & — & 84.0 & 91.5 & 91.7 \\ ImageNet & SL & 88.1 & 96.1 & 100.0 \\ SimCLR & SFSL & 88.1 & 95.1 & 100.0 \\ MoCo & SFSL & 86.4 & 95.3 & 100.0 \\ FractalDB1k & FDSL & 87.4 & 94.9 & 100.0 \\ MV-FractalDB1k & FDSL & 87.6 & 95.7 & 100.0 \\ \hline \end{tabular}
\end{table}
Table 1: Performance in terms of %age accuracy on the ModelNet [109] and the MIRO [110, 111] datasets, respectively [108]. (Note: SFSL = self-supervised learning, SL = supervised learning).  
\begin{table}
\begin{tabular}{l c c c c} \hline
**Dataset** & **Cifar-10** & **Cifar-100** & **Cars** & **Flowers** \\ \hline — & 78.3 & 57.7 & 11.6 & 77.1 \\ PerlinNoiseDB & 94.5 & 77.8 & 62.3 & 96.1 \\ BeizerCurveDB & 96.7 & 80.3 & 82.8 & 98.5 \\ FractalDB1k & 96.8 & 81.6 & 86.0 & 98.3 \\ \hline \end{tabular}
\end{table}
Table 2: Performance comparison of visual transformer pretraining in terms of % age accuracy with FractalDB1k and other FDSL datasets for BezierCurveDB and PerlinNoiseDB [77].