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1. Counterexamples in scale calculus

   https://doi.org/10.1073/pnas.1811701116  

   Filippenko, Benjamin; Zhou, Zhengyi; Wehrheim, Katrin (April 2019, Proceedings of the National Academy of Sciences)

   We construct counterexamples to classical calculus facts such as the inverse and implicit function theorems in scale calculus—a generalization of multivariable calculus to infinite-dimensional vector spaces, in which the reparameterization maps relevant to symplectic geometry are smooth. Scale calculus is a corner stone of polyfold theory, which was introduced by Hofer, Wysocki, and Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves. We show how the novel nonlinear scale-Fredholm notion in polyfold theory overcomes the lack of implicit function theorems, by formally establishing an often implicitly used fact: The differentials of basic germs—the local models for scale-Fredholm maps—vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of polyfold theory. 

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2. Geographic Boosting Tree: Modeling Non-Stationary Spatial Data

   https://doi.org/10.1109/ICMLA51294.2020.00190  

   Deng, Liangdong; Adjouadi, Malek; Rishe, Naphtali (December 2020, 2020 19th IEEE International Conference on Machine Learning and Applications (ICMLA))

   null (Ed.)

   Non-stationarity is often observed in Geographic datasets. One way to explain non-stationarity is to think of it as a hidden "local knowledge" that varies across space. It is inherently difficult to model such data as models built for one region do not necessarily fit another area as the local knowledge could be different. A solution for this problem is to construct multiple local models at various locations, with each local model accounting for a sub-region within which the data remains relatively stationary. However, this approach is sensitive to the size of data, as the local models are only trained from a subset of observations from a particular region. In this paper, we present a novel approach that addresses this problem by aggregating spatially similar sub-regions into relatively large partitions. Our insight is that although local knowledge shifts over space, it is possible for multiple regions to share the same local knowledge. Data from these regions can be aggregated to train a more accurate model. Experiments show that this method can handle non-stationary and outperforms when the dataset is relatively small. 

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3. LVRF: A Latent Variable Based Approach for Exploring Geographic Datasets

   https://doi.org/10.58245/ipsi.tir.2302.02  

   Deng, Liangdong; Mahara, Arpan; Rishe, Naphtali; Adjouadi, Malek (July 2023, IPSI Transactions on Internet Research)

   Geographic datasets are usually accompanied by spatial non-stationarity – a phenomenon that the relationship between features varies across space. Naturally, nonstationarity can be interpreted as the underlying rule that decides how data are generated and alters over space. Therefore, traditional machine learning algorithms are not suitable for handling non-stationary geographic datasets, as they only render a single global model. To solve this problem, researchers often adopt the multiple-local-model approach, which uses different models to account for different sub-regions of space. This approach has been proven efficient but not optimal, as it is inherently difficult to decide the size of subregions. Additionally, the fact that local models are only trained on a subset of data also limits their potential. This paper proposes an entirely different strategy that interprets nonstationarity as a lack of data and addresses it by introducing latent variables to the original dataset. Backpropagation is then used to find the best values for these latent variables. Experiments show that this method is at least as efficient as multiple-local-model-based approaches and has even greater potential. 

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4. Inverse Distance Weighted Random Forests: Modeling Unevenly Distributed Non-Stationary Geographic Data

   https://doi.org/10.1109/ICACSIS51025.2020.9263208  

   Deng, Liangdong; Adjouadi, Malek; Rishe, Naphtali (October 2020, 2020 International Conference on Advanced Computer Science and Information Systems (ICACSIS))

   null (Ed.)

   Recent years saw explosive growth of Human Geography Data, in which spatial non-stationarity is often observed, i.e., relationships between features depend on the location. For these datasets, a single global model cannot accurately describe the relationships among features that vary across space. To address this problem, a viable solution- that has been adopted by many studies-is to create multiple local models instead of a global one, with each local model representing a subregion of the space. However, the challenge with this approach is that the local models are only fitted to nearby observations. For sparsely sampled regions, the data could be too few to generate any high-quality model. This is especially true for Human Geography datasets, as human activities tend to cluster at a few locations. In this paper, we present a modeling method that addresses this problem by letting local models operate within relatively large subregions, where overlapping is allowed. Results from all local models are then fused using an inverse distance weighted approach, to minimize the impact brought by overlapping. Experiments showed that this method handles non-stationary geographic data very Well, even When they are unevenly distributed. 

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5. Bergman-Harmonic Functions on Classical Domains

   https://doi.org/10.1093/imrn/rnz260  

   Xiao, Ming (December 2019, International Mathematics Research Notices)

   Abstract We study Bergman-harmonic functions on classical domains from a new point of view in this paper. We first establish a boundary pluriharmonicity result for Bergman-harmonic functions on classical domains: a Bergman-harmonic function $$u$$ on a classical domain $$D$$ must be pluriharmonic on germs of complex manifolds in the boundary of $$D$$ if $$u$$ has some appropriate boundary regularity. Next we give a new characterization of pluriharmonicity on classical domains which may shed a new light on future study of Bergman-harmonic functions. We also prove characterization results for Bergman-harmonic functions on type I domains. 

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