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1. Investigating the link between impulsivity and obesity through urban scaling laws

   https://doi.org/10.1371/journal.pcsy.0000046  

   Gan, Tian; Succar, Rayan; Macrì, Simone; Porfiri, Maurizio (May 2025, PLOS Complex Systems)

   Oliveira, Marcos (Ed.)

   Impulsivity has been proposed as a key driver of obesity. However, evidence linking impulsivity and obesity has relied on the study of individual factors, with limited account for the urban attributes of obesogenic environments. Here, we investigate the relationship between obesity and impulsivity through urban scaling and causal discovery. For 915 cities in the United States of America, we study the prevalence of obesity in adults, attention deficit hyperactivity disorder (ADHD) in children, and relevant urban features. We observe sublinear scaling of obesity and ADHD with population size, these disorders being less prevalent in larger cities. By applying a causal discovery tool to the deviations of cities from the urban scaling laws, we identify an influence of ADHD on obesity, moderated by lifestyle. The strength of these associations is confirmed by individual-level data on a cohort of 19,333 children, wherein we observe that ADHD modulates obesity both directly and indirectly. 

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2. Scaling of urban income inequality in the USA

   https://doi.org/10.1098/rsif.2021.0223  

   Heinrich Mora, Elisa; Heine, Cate; Jackson, Jacob J.; West, Geoffrey B.; Yang, Vicky Chuqiao; Kempes, Christopher P. (August 2021, Journal of The Royal Society Interface)

   Urban scaling analysis, the study of how aggregated urban features vary with the population of an urban area, provides a promising framework for discovering commonalities across cities and uncovering dynamics shared by cities across time and space. Here, we use the urban scaling framework to study an important, but under-explored feature in this community—income inequality. We propose a new method to study the scaling of income distributions by analysing total income scaling in population percentiles. We show that income in the least wealthy decile (10%) scales close to linearly with city population, while income in the most wealthy decile scale with a significantly superlinear exponent. In contrast to the superlinear scaling of total income with city population, this decile scaling illustrates that the benefits of larger cities are increasingly unequally distributed. For the poorest income deciles, cities have no positive effect over the null expectation of a linear increase. We repeat our analysis after adjusting income by housing cost, and find similar results. We then further analyse the shapes of income distributions. First, we find that mean, variance, skewness and kurtosis of income distributions all increase with city size. Second, the Kullback–Leibler divergence between a city’s income distribution and that of the largest city decreases with city population, suggesting the overall shape of income distribution shifts with city population. As most urban scaling theories consider densifying interactions within cities as the fundamental process leading to the superlinear increase of many features, our results suggest this effect is only seen in the upper deciles of the cities. Our finding encourages future work to consider heterogeneous models of interactions to form a more coherent understanding of urban scaling. 

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3. Linear scaling between microbial predator and prey densities in the global ocean

   https://doi.org/10.1111/1462-2920.16274  

   Rajakaruna, Harshana; Omta, Anne_Willem; Carr, Eric; Talmy, David (November 2022, Environmental Microbiology)

   Abstract It has been proposed that microbial predator and prey densities are related through sublinear power laws. We revisited previously published biomass and abundance data and fitted Power‐law Biomass Scaling Relationships (PBSRs) between marine microzooplankton predators (Z) and phytoplankton prey (P), and marine viral predators (V) and bacterial prey (B). We analysed them assuming an error structure given by Type II regression models which, in contrast to the conventional Type I regression model, accounts for errors in both the independent and the dependent variables. We found that the data support linear relationships, in contrast to the sublinear relationships reported by previous authors. The scaling exponent yields an expected value of 1 with some spread in different datasets that was well‐described with a Gaussian distribution. Our results suggest that the ratiosZ/P, andV/Bare on average invariant, in contrast to the hypothesis that they systematically decrease with increasingPand B, respectively, as previously thought. 

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4. Understanding Scaling Laws with Statistical and Approximation Theory for Transformer Neural Networks on Intrinsically Low-dimensional Data

   Havrilla, Alex; Liao, Wenjing (January 2025, Advances in Neural Information Processing Systems 37 (NeurIPS 2024) Main Conference Track)

   When training deep neural networks, a model's generalization error is often observed to follow a power scaling law dependent both on the model size and the data size. Perhaps the best known example of such scaling laws are for transformer-based large language models (**LLMs**), where networks with billions of parameters are trained on trillions of tokens of text. Yet, despite sustained widespread interest, a rigorous understanding of why transformer scaling laws exist is still missing. To answer this question, we establish novel statistical estimation and mathematical approximation theories for transformers when the input data are concentrated on a low-dimensional manifold. Our theory predicts a power law between the generalization error and both the training data size and the network size for transformers, where the power depends on the intrinsic dimension d of the training data. Notably, the constructed model architecture is shallow, requiring only logarithmic depth in d. By leveraging low-dimensional data structures under a manifold hypothesis, we are able to explain transformer scaling laws in a way which respects the data geometry. Moreover, we test our theory with empirical observation by training LLMs on natural language datasets. We find the observed empirical scaling laws closely agree with our theoretical predictions. Taken together, these results rigorously show the intrinsic dimension of data to be a crucial quantity affecting transformer scaling laws in both theory and practice. 

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5. Non-asymptotic superlinear convergence of standard quasi-Newton methods

   https://doi.org/10.1007/s10107-022-01887-4  

   Jin, Qiujiang; Mokhtari, Aryan (September 2022, Mathematical Programming)

   Abstract In this paper, we study and prove the non-asymptotic superlinear convergence rate of the Broyden class of quasi-Newton algorithms which includes the Davidon–Fletcher–Powell (DFP) method and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. The asymptotic superlinear convergence rate of these quasi-Newton methods has been extensively studied in the literature, but their explicit finite–time local convergence rate is not fully investigated. In this paper, we provide a finite–time (non-asymptotic) convergence analysis for Broyden quasi-Newton algorithms under the assumptions that the objective function is strongly convex, its gradient is Lipschitz continuous, and its Hessian is Lipschitz continuous at the optimal solution. We show that in a local neighborhood of the optimal solution, the iterates generated by both DFP and BFGS converge to the optimal solution at a superlinear rate of$$(1/k)^{k/2}$$ ${(1/k)}^{k/2}$, wherekis the number of iterations. We also prove a similar local superlinear convergence result holds for the case that the objective function is self-concordant. Numerical experiments on several datasets confirm our explicit convergence rate bounds. Our theoretical guarantee is one of the first results that provide a non-asymptotic superlinear convergence rate for quasi-Newton methods. 

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