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.

The key quantity of interest here is the Riemannian metric, which characterizes the Riemannian geometry and defines straight lines and derivatives on the manifold.

We report the details of hyperparameters in our experiments.

Do the main claims made in the abstract and introduction accurately reflect the paper's If you ran experiments (e.g. for benchmarks)... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Y es] See A.2 (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? Did you include the total amount of compute and the type of resources used (e.g., type Did you include any new assets either in the supplemental material or as a URL? [Y es] Did you discuss whether and how consent was obtained from people whose data you're If you used crowdsourcing or conducted research with human subjects... (a) For a detailed description and intended uses, please refer to 1. A.2 Dataset Accessibility We plan to host and maintain this dataset on HuggingFace. A.4 Dataset Examples Example question-answer pairs are provided in Tables 9 10 11, . Example Question "What does the symbol mean in Equation 1?" Answer "The symbol in Equation 1 represents "follows this distribution". "Can you provide more information about what is meant by'generative process in "The generative process refers to Eq. (2), which is a conceptual equation representing Question "How does the DeepMoD method differ from what is written in/after Eq 3?" Answer "We add noise only to Question "How to do the adaptive attack based on Eq.(16)? "By Maximizing the loss in Eq (16) using an iterative method such as PGD on the end-to-end model we attempt to maximize the loss to cause misclassification while Question "How does the proposed method handle the imputed reward?" "The proposed method uses the imputed reward in the second part of Equation 1, "Table 2 is used to provide a comparison of the computational complexity of the "Optimal number of clusters affected by the number of classes or similarity between "The authors have addressed this concern by including a new experiment in Table 4 of Question "Can you clarify the values represented in Table 1?" Answer "The values in Table 1 represent the number of evasions, which shows the attack "The experiments in table 1 do not seem to favor the proposed method much; softmax Can the authors explain why this might be the case?" Answer "The proposed method reduces to empirical risk minimization with a proper loss, and However, the authors hope that addressing concerns about the method's theoretical Question "Does the first row of Table 2 correspond to the offline method?"

Forsurfaceplotsof ResNet-56, see Figure 1.

Findthesmallest0 (n, k, ), where (n, k, )= c1 r logn+ log ( 1/ ) k . Then, withprobabilityatleast1 , theresultingclassifiergn satisfiesthefollowing: foreverypointx 2 supp(µ), if n C adv (x) max log 1 adv (x) , log 1 thengn(x)= g (x).

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