## 2 Materials and Methods
### Feature Reduction and Main Clustering
#### 2.2.1 Convolutional Autoencoder  
Since their invention in the 1980s, autoencoders have successfully been applied to a huge variety of different tasks in computational research and engineering primarily focusing on dimensionality reduction and information retrieval [43; 44; 45; 46]. They have been developed to find a feature representation (encoding) for a set of (high-dimensional) data by modeling the manifold of a dataset in a selfsupervised manner while ignoring noise. In order to maintain the local structure of certain data, which is particularly important in the present case to be able to identify potential correlations between different light spectra, [PERSON] et al. [47] proposed to extend the autoencoder approach by a convolutional architecture, where corresponding weights are shared among all locations in the input for preserving spatial locality. To prepare the spectral data to fit the convolutional autoencoder (CAE) architecture, the corresponding \(1\times 401\) parameter vectors, one for each input light spectrum, are cropped and re-arranged in \(20\times 20\) dimensional arrays. For this purpose, the original wavelength range was truncated to 380-779 nm. The error introduced by this truncation is negligible with regard to lighting applications as the human visual system is almost completely insensitive to wavelengths larger than 750 nm. The \(20\times 20\) dimensional arrays can then be interpreted as grey-scale pictures, where each pixel value represents the assigned wavelength's light level. By applying the CAE to the set of input data considered in this work, a significant feature reduction from \(20\times 20\) to 15 parameters could be achieved. As can be seen from Figure 4, the CAE consists of an encoding and decoding stage. During encoding, the CAE tries to find an appropriate, dimensionally reduced feature representation of the input light spectra. During decoding, this feature representation is then used to reconstruct the original \(20\times 20\) data. This allows the CAE to calculate the error between the input and output/reconstructed light spectra to optimize the internal layers with the aim of minimizing the error. Figure 3: Algorithmic structure of the physical unbiased light clustering. Figure 4: Illustration of the convolutional autoencoder used to process the spectral input data. Before processing, the light spectra are cropped to 380–779 nm and re-arranged as \(20\times 20\) dimensional data arrays. Subsequent encoding and decoding, while minimizing the corresponding loss function between in- and output data, resulted in a set of optimized \(1\times 15\) dimensional encoded feature vectors representing the original input light spectra. A pre-processing step with a normalizing feature scaling between zero to one was applied to limit the range of the input data and to find the minimum (min) and maximum (max) of all possible input spectra. In this work, a kernel_size of 3 was used. The kernel_size defines the window area in which the data points are processed. With the spectral input data being represented as \(20\times 20\) arrays, the kernel windows applied to this array structure always include wavelengths from three non-contiguous regions of the spectrum. This results in the creation of a more generalized convolutional layer. The used autoencoder basically combines two of such layers. The first one increases the dimension of the input data arrays to \(10\times 18\times 18\), where each additional dimension represents a weighted kernel filtering of the data to learn the complexity of input spectra [43; 44; 45]. The second convolutional layer then reduces the data to \(4\times 7\times 7\) dimensions after a \(2\times 2\) max-pooling layer [48]. This reduction of the amount of filters was performed to reduce the computational cost for the next fully connected layers. Afterwards, the fully connected linear layers are used to further reduce the dimensionality to come up with the final encoding of the light spectra in terms of \(1\times 15\) dimensional encoded feature vectors.