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1. Parabolic Regularity of Spectral Functions

   https://doi.org/10.1287/moor.2023.0010  

   Mohammadi, Ashkan; Sarabi, Ebrahim (August 2024, Mathematics of Operations Research)

   This paper is devoted to the study of the second-order variational analysis of spectral functions. It is well-known that spectral functions can be expressed as a composite function of symmetric functions and eigenvalue functions. We establish several second-order properties of spectral functions when their associated symmetric functions enjoy these properties. Our main attention is given to characterize parabolic regularity for this class of functions. It was observed recently that parabolic regularity can play a central rule in ensuring the validity of important second-order variational properties, such as twice epi-differentiability. We demonstrates that for convex spectral functions, their parabolic regularity amounts to that of their symmetric functions. As an important consequence, we calculate the second subderivative of convex spectral functions, which allows us to establish second-order optimality conditions for a class of matrix optimization problems. 

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2. Backward transfer, the relationship between new learning and prior ways of reasoning, and action versus process views of linear functions

   https://doi.org/10.1080/10986065.2022.2037043  

   Hohensee, Charles; Willoughby, Laura; Gartland, Sara (February 2022, Mathematical Thinking and Learning)

   Backward transfer is defined as the influence that new learning has on individuals’ prior ways of reasoning. In this article, we report on an exploratory study that examined the influences that quadratic functions instruction in real classrooms had on students’ prior ways of reasoning about linear functions. Two algebra classes and their teachers at two comprehensive high schools served as the participants. Both schools drew from lowsocioeconomic urban populations. The study involved paper-and-pencil assessments about linear functions that were administered before and after a four- to five-week instructional unit on quadratic functions. The teachers were instructed to teach the quadratic functions unit using their regular approach. Qualitative analysis revealed three kinds of backward transfer influences and each influence was related to a shift in how the students reasoned about functions in terms of an action or process view of functions. Additionally, features of the instruction in each class provided plausible explanations for the similarities and differences in backward transfer effects across the two classrooms. These results offer insights into backward transfer, the relationship between prior knowledge and new learning, aspects of reasoning about linear functions, and instructional approaches to teaching functions. 

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3. Structural diversity as a predictor of ecosystem function

   https://doi.org/10.1088/1748-9326/ab49bb  

   LaRue, Elizabeth A.; Hardiman, Brady S.; Elliott, Jessica M.; Fei, Songlin (October 2019, Environmental Research Letters)

   Abstract Biodiversity is believed to be closely related to ecosystem functions. However, the ability of existing biodiversity measures, such as species richness and phylogenetic diversity, to predict ecosystem functions remains elusive. Here, we propose a new vector of diversity metrics, structural diversity, which directly incorporates niche space in measuring ecosystem structure. We hypothesize that structural diversity will provide better predictive ability of key ecosystem functions than traditional biodiversity measures. Using the new lidar-derived canopy structural diversity metrics on 19 National Ecological Observation Network forested sites across the USA, we show that structural diversity is a better predictor of key ecosystem functions, such as productivity, energy, and nutrient dynamics than existing biodiversity measures (i.e. species richness and phylogenetic diversity). Similar to existing biodiversity measures, we found that the relationships between structural diversity and ecosystem functions are sensitive to environmental context. Our study indicates that structural diversity may be as good or a better predictor of ecosystem functions than species richness and phylogenetic diversity. 

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4. Log orthogonal functions: approximation properties and applications

   https://doi.org/10.1093/imanum/draa087  

   Chen, Sheng; Shen, Jie (December 2020, IMA Journal of Numerical Analysis)

   Abstract We present two new classes of orthogonal functions, log orthogonal functions and generalized log orthogonal functions, which are constructed by applying a $$\log $$ mapping to Laguerre polynomials. We develop basic approximation theory for these new orthogonal functions, and apply them to solve several typical fractional differential equations whose solutions exhibit weak singularities. Our error analysis and numerical results show that our methods based on the new orthogonal functions are particularly suitable for functions that have weak singularities at one endpoint and can lead to exponential convergence rate, as opposed to low algebraic rates if usual orthogonal polynomials are used. 

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5. Neural Sculpting: Uncovering hierarchically modular task structure in neural networks through pruning and network analysis

   Shreyas Malakarjun Patil, Loizos Michael (December 2023, Advances in neural information processing systems)

   Natural target functions and tasks typically exhibit hierarchical modularity – they can be broken down into simpler sub-functions that are organized in a hierarchy. Such sub-functions have two important features: they have a distinct set of inputs (input-separability) and they are reused as inputs higher in the hierarchy (reusability). Previous studies have established that hierarchically modular neural networks, which are inherently sparse, offer benefits such as learning efficiency, generalization, multi-task learning, and transfer. However, identifying the underlying sub-functions and their hierarchical structure for a given task can be challenging. The high-level question in this work is: if we learn a task using a sufficiently deep neural network, how can we uncover the underlying hierarchy of sub-functions in that task? As a starting point, we examine the domain of Boolean functions, where it is easier to determine whether a task is hierarchically modular. We propose an approach based on iterative unit and edge pruning (during training), combined with network analysis for module detection and hierarchy inference. Finally, we demonstrate that this method can uncover the hierarchical modularity of a wide range of Boolean functions and two vision tasks based on the MNIST digits dataset. 

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