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The first term of its optimization objective depicts the shrinking of the data-encompassing hypersphere around the given center $\mathbf{c}$. For each data sample $\{\mathbf{x}_1, \dots, \mathbf{x}_n\}$, its geometric distance to $\mathbf{c}$ in the latent space produced by the neural network $\phi(\wc; \mathcal{W})$ is minimized proportionally to the amount of data samples $n$. The second term is a standard L2 regularization term which prevents overfitting with hyperparameter $\lambda > 0$ and $\|\wc\|_F$ denoting the Frobenius norm. -\citeauthor{deepsad} argue that the pre-training step employing an autoencoder—originally introduced in Deep SVDD—not only allows a geometric interpretation of the method as minimum volume estimation i.e., the shrinking of the data encompassing hypersphere but also a probabilistic one as entropy minimization over the latent distribution. The autoencoding objective during pre-training implicitly maximizes the mutual information between the data and its latent representation, aligning the approach with the Infomax principle while encouraging a latent space with minimal entropy. This insight enabled \citeauthor{deepsad} to introduce an additional term in DeepSAD’s objective, beyond that of its predecessor Deep SVDD, which incorporates labeled data to better capture the characteristics of normal and anomalous data. They demonstrate that DeepSAD’s objective effectively models the latent distribution of normal data as having low entropy, while that of anomalous data is characterized by higher entropy. In this framework, anomalies are interpreted as being generated from an infinite mixture of distributions that differ from the normal data distribution. +\citeauthor{deepsad} argue that the pre-training step employing an autoencoder—originally introduced in Deep SVDD—not only allows a geometric interpretation of the method as minimum volume estimation i.e., the shrinking of the data encompassing hypersphere but also a probabilistic one as entropy minimization over the latent distribution. The autoencoding objective during pre-training implicitly maximizes the mutual information between the data and its latent representation, aligning the approach with the Infomax principle while encouraging a latent space with minimal entropy. This insight enabled \citeauthor{deepsad} to introduce an additional term in DeepSAD’s objective, beyond that of its predecessor Deep SVDD, which incorporates labeled data to better capture the characteristics of normal and anomalous data. They demonstrate that DeepSAD’s objective effectively models the latent distribution of normal data as having low entropy, while that of anomalous data is characterized by higher entropy. In this framework, anomalies are interpreted as being generated from an infinite mixture of distributions that differ from the normal data distribution. The introduction of this aforementioned term in DeepSAD's objective allows it to learn in a semi-supervised way, which helps the model better position known normal samples near the hypersphere center and push known anomalies farther away, thereby enhancing its ability to differentiate between normal and anomalous data. -The introduction of the aforementioned term in Deep SAD's objective allows it to learn in a semi-supervised way, though it can operate in a fully unsupervised mode—effectively reverting to its predecessor, Deep SVDD—when no labeled data are available. This additional supervision helps the model better position known normal samples near the hypersphere center and push known anomalies farther away, thereby enhancing its ability to differentiate between normal and anomalous data. - - -From \ref{eq:deepsvdd_optimization_objective} it is easy to understand Deep SAD's optimization objective seen in \ref{eq:deepsad_optimization_objective} which additionally defines $m$ number of labeled data samples $\{(\mathbf{\tilde{x}}_1, \tilde{y}_1), \dots, (\mathbf{\tilde{x}}_m, \tilde{y}_1)\} \in \mathcal{X} \times \mathcal{Y}$ and $\mathcal{Y} = \{-1,+1\}$ for which $\tilde{y} = +1$ denotes normal and $\tilde{y} = -1$ anomalous samples as well as a new hyperparameter $\eta > 0$ which can be used to balance the strength with which labeled and unlabeled samples contribute to the training. +From equation~\ref{eq:deepsvdd_optimization_objective} it is easy to understand DeepSAD's optimization objective seen in equation~\ref{eq:deepsad_optimization_objective} which additionally defines $m$ number of labeled data samples $\{(\mathbf{\tilde{x}}_1, \tilde{y}_1), \dots, (\mathbf{\tilde{x}}_m, \tilde{y}_1)\} \in \mathcal{X} \times \mathcal{Y}$ and $\mathcal{Y} = \{-1,+1\}$ for which $\tilde{y} = +1$ denotes normal and $\tilde{y} = -1$ anomalous samples as well as a new hyperparameter $\eta > 0$ which can be used to balance the strength with which labeled and unlabeled samples contribute to the training. \begin{equation} \label{eq:deepsad_optimization_objective} @@ -392,7 +386,7 @@ From \ref{eq:deepsvdd_optimization_objective} it is easy to understand Deep SAD' +\frac{\lambda}{2}\sum_{\ell=1}^{L}\|\mathbf{W}^{\ell}\|_{F}^{2}. \end{equation} -The first term of \ref{eq:deepsad_optimization_objective} stays mostly the same, differing only in its consideration of the introduced $m$ labeled datasamples for its proportionality. The second term is newly introduced to incorporate the labeled data samples with hyperparameter $\eta$'s strength, by either minimizing or maximizing the distance between the samples latent represenation and $\mathbf{c}$ depending on each data samples label $\tilde{y}$. The third term, is kept identical compared to Deep SVDD as standard L2 regularization. It can also be observed that in case of $m = 0$ labeled samples, Deep SAD falls back to the same optimization objective of Deep SVDD and can therefore be used in a completely unsupervised fashion as well. +The first term of equation~\ref{eq:deepsad_optimization_objective} stays mostly the same, differing only in its consideration of the introduced $m$ labeled datasamples for its proportionality. The second term is newly introduced to incorporate the labeled data samples with hyperparameter $\eta$'s strength, by either minimizing or maximizing the distance between the samples latent represenation and $\mathbf{c}$ depending on each data samples label $\tilde{y}$. The standard L2 regularization is kept identical to Deep SVDD's optimization objective. It can also be observed that in case of $m = 0$ labeled samples, DeepSAD falls back to Deep SVDD's optimization objective and can therefore be used in a completely unsupervised fashion as well. \newsubsubsectionNoTOC{Hyperparameters} @@ -525,7 +519,7 @@ Taken together, the percentage of missing points and the proportion of near-sens \newsection{preprocessing}{Preprocessing Steps and Labeling} -As described in Section~\ref{sec:algorithm_description}, the method under evaluation is data type agnostic and can be adapted to work with any kind of data by choosing a suitable autoencoder architecture. In our case, the input data are point clouds produced by a lidar sensor. Each point cloud contains up to 65,536 points, with each point represented by its \emph{X}, \emph{Y}, and \emph{Z} coordinates. To tailor the Deep SAD architecture to this specific data type, we would need to design an autoencoder suitable for processing three-dimensional point clouds. Although autoencoders can be developed for various data types, \citetitle{autoencoder_survey}~\cite{autoencoder_survey} noted that over 60\% of recent research on autoencoders focuses on two-dimensional image classification and reconstruction. Consequently, there is a more established understanding of autoencoder architectures for images compared to those for three-dimensional point clouds. +As described in Section~\ref{sec:algorithm_description}, the method under evaluation is data type agnostic and can be adapted to work with any kind of data by choosing a suitable autoencoder architecture. In our case, the input data are point clouds produced by a lidar sensor. Each point cloud contains up to 65,536 points, with each point represented by its \emph{X}, \emph{Y}, and \emph{Z} coordinates. To tailor the DeepSAD architecture to this specific data type, we would need to design an autoencoder suitable for processing three-dimensional point clouds. Although autoencoders can be developed for various data types, \citetitle{autoencoder_survey}~\cite{autoencoder_survey} noted that over 60\% of recent research on autoencoders focuses on two-dimensional image classification and reconstruction. Consequently, there is a more established understanding of autoencoder architectures for images compared to those for three-dimensional point clouds. For this reason and to simplify the architecture, we converted the point clouds into two-dimensional grayscale images using a spherical projection. This approach—proven sucessful in related work~\cite{degradation_quantification_rain}—encodes each lidar measurement as a single pixel, where the pixel’s grayscale value is determined by the reciprocal range, calculated as $v = \frac{1}{\sqrt{\emph{X}^2 + \emph{Y}^2 + \emph{Z}^2}}$. Given the lidar sensor's configuration, the resulting images have a resolution of 2048 pixels in width and 32 pixels in height. Missing measurements in the point cloud are mapped to pixels with a brightness value of $v = 0$.