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   Su, Jingtong; Zhang, YaShi; Tsilivis, Nikolaos; Kempe, Julia (April 2024, Transactions on Machine Learning Research)

   Neural Collapse refers to the curious phenomenon in the end of training of a neural network, where feature vectors and classification weights converge to a very simple geometrical arrangement (a simplex). While it has been observed empirically in various cases and has been theoretically motivated, its connection with crucial properties of neural networks, like their generalization and robustness, remains unclear. In this work, we study the stability properties of these simplices. We find that the simplex structure disappears under small adversarial attacks, and that perturbed examples "leap" between simplex vertices. We further analyze the geometry of networks that are optimized to be robust against adversarial perturbations of the input, and find that Neural Collapse is a pervasive phenomenon in these cases as well, with clean and perturbed representations forming aligned simplices, and giving rise to a robust simple nearest-neighbor classifier. By studying the propagation of the amount of collapse inside the network, we identify novel properties of both robust and non-robust machine learning models, and show that earlier, unlike later layers maintain reliable simplices on perturbed data. 

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2. Self-Collapse Lithography

   https://doi.org/10.1021/acs.nanolett.7b02269  

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4. Limited diffusion of scientific knowledge forecasts collapse

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5. Carbon, Climate, and Collapse: Coupling Climate Feedbacks and Resource Dynamics to Predict Societal Collapse

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   Savitsky, Greta; Burnett, Grace; Beckage, Brian (September 2025, Systems)

   Anthropogenic climate change threatens production of essential natural resources, including food, fiber, and fresh water, and provisioning of ecosystem services such as carbon sequestration, increasing the risk of societal collapse. The Human and Nature Dynamics (HANDY) model simulates the effect of resource overexploitation on societal collapse but lacks representation of feedbacks between climate change and resource regeneration in ecological systems. We extend the HANDY model by integrating models of climate change and ecological function to examine the risk of societal collapse. We conducted a sensitivity analysis of our expanded model by systematically varying key parameters to examine the range of plausible socio-ecological conditions and evaluate model uncertainty. We find that lowered greenhouse gas emissions and resilient ecosystems can delay societal collapse by up to approximately 500 years, but that any scenario with greater than net-zero greenhouse gas emissions ultimately leads to societal collapse driven by climate-induced loss of ecosystem function. Reductions in greenhouse gas emissions are the most effective intervention to delay or prevent societal collapse, followed by the conservation and management of resilient ecological systems to sequester atmospheric carbon. 

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