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.

Recently, Forré (arXiv:2104.11547, 2021) introduced transitional conditional independence, a notion of conditional independence that provides a unified framework for both random and non-stochastic variables. The original paper establishes a strong global Markov property connecting transitional conditional independencies with suitable graphical separation criteria for directed mixed graphs with input nodes (iDMGs), together with a version of causal calculus for iDMGs in a general measure-theoretic setting. These notes aim to further illustrate the motivations behind this framework and its connections to the literature, highlight certain subtlies in the general measure-theoretic causal calculus, and extend the "one-line" formulation of the ID algorithm of Richardson et al. (Ann. Statist. 51(1):334--361, 2023) to the general measure-theoretic setting.

Recursive partitioning methods provide computationally efficient surrogates for the Wasserstein distance, yet their statistical behavior and their resolution in the small-discrepancy regime remain insufficiently understood. We study Recursive Rank Matching (RRM) as a representative instance of this class under a population-anchored reference. In this setting, we establish consistency and an explicit convergence rate for the anchored empirical RRM under the quadratic cost. We then identify a dominant mismatch mechanism responsible for the loss of resolution in the small-discrepancy regime. Based on this analysis, we introduce Selective Recursive Rank Matching (SRRM), which suppresses the resulting dominant mismatches and yields a higher-fidelity practical surrogate for the Wasserstein distance at moderate additional computational cost.

This paper develops new variance-reduction techniques for the forward-reflected-backward splitting (FRBS) method to solve a class of possibly nonmonotone stochastic composite inclusions. Unlike unbiased estimators such as mini-batching, developing stochastic biased variants faces a fundamental technical challenge and has not been utilized before for inclusions and fixed-point problems. We fill this gap by designing a new framework that can handle both unbiased and biased estimators. Our main idea is to construct stochastic variance-reduced estimators for the forward-reflected direction and use them to perform iterate updates. First, we propose a class of unbiased variance-reduced estimators and show that increasing mini-batch SGD, loopless-SVRG, and SAGA estimators fall within this class. For these unbiased estimators, we establish a $\mathcal{O}(1/k)$ best-iterate convergence rate for the expected squared residual norm, together with almost-sure convergence of the iterate sequence to a solution. Consequently, we prove that the best oracle complexities for the $n$-finite-sum and expectation settings are $\mathcal{O}(n^{2/3}ε^{-2})$ and $\mathcal{O}(ε^{-10/3})$, respectively, when employing loopless-SVRG or SAGA, where $ε$ is a desired accuracy. Second, we introduce a new class of biased variance-reduced estimators for the forward-reflected direction, which includes SARAH, Hybrid SGD, and Hybrid SVRG as special instances. While the convergence rates remain valid for these biased estimators, the resulting oracle complexities are $\mathcal{O}(n^{3/4}ε^{-2})$ and $\mathcal{O}(ε^{-5})$ for the $n$-finite-sum and expectation settings, respectively. Finally, we conduct two numerical experiments on AUC optimization for imbalanced classification and policy evaluation in reinforcement learning.

However, the problem of provably editing a DNN to satisfy a property remains challenging.

Further taking the usual assumption that X is compact. Let us start with Proposition 3, a central observation needed in Theorem 2. Put into words Now, we can proceed to prove the universality part of Theorem 2. Since the task admits a smooth separator, By Fubini's theorem and Proposition 3, we have F The reader can think of λ as a uniform distribution over G. (as in Theorem 2). The result follows directly from the combination of de Finetti's theorem [ Combining this with Kallenberg's noise transfer theorem we have that the weights and Assumption 1 or ii) is an inner-product decision graph problem as in Definition 3. Further, the task has infinitely (as in Theorem 2). Finally, we follow Proposition 2's proof by simply replacing de Finetti's with Aldous-Hoover's theorem. Define an RLC that samples the linear coefficients as follows.