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1. The geometry of adversarial training in binary classification

   https://doi.org/10.1093/imaiai/iaac029  

   Bungert, Leon; García Trillos, Nicolás; Murray, Ryan (February 2023, Information and Inference: A Journal of the IMA)

   Abstract We establish an equivalence between a family of adversarial training problems for non-parametric binary classification and a family of regularized risk minimization problems where the regularizer is a nonlocal perimeter functional. The resulting regularized risk minimization problems admit exact convex relaxations of the type $$L^1+\text{(nonlocal)}\operatorname{TV}$$, a form frequently studied in image analysis and graph-based learning. A rich geometric structure is revealed by this reformulation which in turn allows us to establish a series of properties of optimal solutions of the original problem, including the existence of minimal and maximal solutions (interpreted in a suitable sense) and the existence of regular solutions (also interpreted in a suitable sense). In addition, we highlight how the connection between adversarial training and perimeter minimization problems provides a novel, directly interpretable, statistical motivation for a family of regularized risk minimization problems involving perimeter/total variation. The majority of our theoretical results are independent of the distance used to define adversarial attacks. 

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2. Nucleation of Fracture: The First-Octant Evidence Against Classical Variational Phase-Field Models

   https://doi.org/10.1115/1.4067146  

   Kamarei, Farhad; Dolbow, John E; Lopez-Pamies, Oscar (January 2025, Journal of Applied Mechanics)

   Abstract As a companion work to [1], this article presents a series of simple formulae and explicit results that illustrate and highlight why classical variational phase-field models cannot possibly predict fracture nucleation in elastic brittle materials. The focus is on “tension-dominated” problems where all principal stresses are nonnegative, that is, problems taking place entirely within the first octant in the space of principal stresses. 

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3. Stochastic blockmodeling for learning the structure of optimization problems

   https://doi.org/10.1002/aic.17415  

   Mitrai, Ilias; Tang, Wentao; Daoutidis, Prodromos (September 2021, AIChE Journal)

   Abstract Decomposition‐based solution algorithms for optimization problems depend on the underlying latent block structure of the problem. Methods for detecting this structure are currently lacking. In this article, we propose stochastic blockmodeling (SBM) as a systematic framework for learning the underlying block structure in generic optimization problems. SBM is a generative graph model in which nodes belong to some blocks and the interconnections among the nodes are stochastically dependent on their block affiliations. Hence, through parametric statistical inference, the interconnection patterns underlying optimization problems can be estimated. For benchmark optimization problems, we show that SBM can reveal the underlying block structure and that the estimated blocks can be used as the basis for decomposition‐based solution algorithms which can reach an optimum or bound estimates in reduced computational time. Finally, we present a general software platform for automated block structure detection and decomposition‐based solution following distributed and hierarchical optimization approaches. 

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4. The Social Risks of Science

   https://doi.org/10.1002/hast.1196  

   Herington, Jonathan; Tanona, Scott (December 2020, Hastings Center Report)

   Abstract Many instances of scientific research impose risks, not just on participants and scientists but also on third parties. This class ofsocial risksunifies a range of problems previously treated as distinct phenomena, including so‐called bystander risks, biosafety concerns arising from gain‐of‐function research, the misuse of the results of dual‐use research, and the harm caused by inductive risks. The standard approach to these problems has been to extend two familiar principles from human subjects research regulations—a favorable risk‐benefit ratio and informed consent. We argue, however, that these moral principles will be difficult to satisfy in the context of widely distributed social risks about which affected parties may reasonably disagree. We propose that framing these risks as political rather than moral problems may offer another way. By borrowing lessons from political philosophy, we propose a framework that unifies our discussion of social risks and the possible solutions to them. 

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5. Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere?

   https://doi.org/10.1007/s11118-024-10186-w  

   Chafaï, Djalil; Matzke, Ryan_W; Saff, Edward_B; Vu, Minh_Quan_H; Womersley, Robert_S (December 2024, Potential Analysis)

   Abstract We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium measure is the uniform distribution on a sphere. We develop general necessary and general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard – Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which a certain dimension reduction phenomenon occurs: the support of the equilibrium measure becomes a sphere. We also briefly discuss the relationship between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk–Hecke formula, and the calculus of hypergeometric functions. 

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