Proposition 2. (From main text) The Bayes error of flow models is monotonically increasing in . That is, for 0 < 0, we have that EBayes(ˆp) EBayes(ˆp 0). B.1 Hardness of Classes In addition to measuring the difficulty of classification tasks relative to one another, it also may be of interest to evaluate the relative difficulty of individual classes within a particular task. A natural way to do this is by looking at the error of one-vs-all classification tasks. The optimal Bayes classifier in this task is CBayes(x)= 0 if logpj(x) logp j(x), 1 otherwise .

In addition to our main results presented in Section 4 of the paper, we also performed various exploratory experiments to investigate further application cases of the METASIN activations. The experiments cover image classification where we show favorable results of using convolutional METASIN networks over baseline RELU networks, as well as various overfitting experiments to explore the use of METASIN activations with MLPs. In all MLP experiments, we use METASIN with K = 10 sine components, and distribute the frequencies evenly across the range [1,35]. The initialization of the remaining parameters follows the description provided in Section 3. Bicycle Figure 5: Visualization of selected reconstructed frames of the video. To fully appreciate the details and visual cues presented in the figure, we recommend visualizing the figures in color and zooming in for a more comprehensive analysis.

Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a way to incorporate topological priors or to regularize machine learning models. This is usually achieved by minimizing adequate, topologically-informed losses based on these descriptors, which, in turn, naturally raises theoretical and practical questions about the possibility of optimizing such loss functions using gradient-based algorithms. This has been an active research field in the topological data analysis community over the last decade, and various techniques have been developed to enable optimization of persistence-based loss functions with gradient descent schemes. This survey presents the current state of this field, covering its theoretical foundations, the algorithmic aspects, and showcasing practical uses in several applications. It includes a detailed introduction to persistence theory and, as such, aims at being accessible to mathematicians and data scientists newcomers to the field. It is accompanied by an open-source library which implements the different approaches covered in this survey, providing a convenient playground for researchers to get familiar with the field.