## 1 Introduction
### Approaches for High-Dimensional Data  
To overcome the limitations of the previously described approaches, when it comes to high-dimensionality problems, ML methods including an estimation of the confidence of the predictions in their output (e.g., Gaussian Processes, ensemble models like Random Forest) can be employed to detect outliers. For example, [37] used an embedded AD approach, based on the probabilities assigned to the output classes of a neural network classifier, as an effective approach to analyze the likelihood of correct predictions. In other words, if the likelihood of correct prediction is low for a molecule, then the latter is likely to be an outlier. Technically, a softmax layer was applied to the network outputs to obtain the probability of each class for each molecule, where the feature space was composed of 1500 to 3700 descriptors approximately. Similarly, in [38], the predictions made by an ensemble of classification decision trees were used to evaluate the confidence of the final prediction for a molecule, and therefore the outlier character of that molecule. More concretely, the fraction of trees that voted for the majority vote was used as the confidence (or probability) of the prediction. The method was applied to problems dealing with a feature space of approximately 100 to 1000 dimensions and showed significant robustness towards the high dimensionality, in comparison to the classical Euclidean distance-based approach. The same methodology can also be implemented via an ensemble of regression trees (or of other regression models), by using the variance of the predictions as a measure of the reliability of the final prediction (i.e., the average of the predictions of the ensemble of trees) [39; 40]. Gaussian Processes (GP) can also be exploited for confidence estimation. In fact, GP defines a class of ML models that are based on the prediction of a posterior distribution of functions, characterized by their mean and variance [41]. This variance can therefore be used as a measure of the reliability of a prediction. In [39], an ensemble of regression trees (RF) and GP models outperformed classical approaches in the definition of the AD. Similarly, in [42], the AD definition via confidence estimation showed better results than the other investigated methods. All these techniques are based on the confidence estimation of ML predictions and on a less binary (i.e., inside/outside AD) approach to dealing with the definition of an AD, in comparison to classical AD methods, by providing a more shaded evaluation of the prediction reliability according to different levels of confidence. This characteristic renders them more compatible with high-dimensional problems. Among the different outlier detection techniques that can be employed in high-dimensional problems, the methods based on projections in lower-dimensional subspaces are also very popular. In [9], a point was considered an outlier if there was a lower-dimensional subspace where this point would be located in a region of abnormally low density. To avoid an exponential search of relevant subspaces, evolutionary algorithms were implemented. Additionally, the density was obtained by dividing the space into grids and then calculating a density coefficient (the sparsity coefficient) for each cell of the grids. Other works used different projection techniques, such as PCA, axis parallel subspace spanned by a point's neighbors or Hilbert space-filling curve, generally followed by a classical distance- or density-based technique, such as kernel density estimation or distance to the \(kth\) nearest neighbor(s), to identify outliers [43; 44; 45; 46; 47]. More specific methods, without projection to a subspace of lower dimension, can also be found in the literature. For example, [48] exploited angles as a measure for detecting outliers, instead of distances, as the latter are very sensitive to high-dimensional data.