**Discrepancy Theory**

In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of classical discrepancy theory, namely distributing points in some space such that they are evenly distributed with respect to some (mostly geometrically defined) subsets. The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one. Discrepancy theory can be described as the study of inevitable irregularities of distributions, in measure-theoretic and combinatorial settings. Just as Ramsey theory elucidates the impossibility of total disorder, discrepancy theory studies the deviations from total uniformity. A significant event in the history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval. … **Zooming Network (ZN)**

Structural information is important in natural language understanding. Although some current neural net-based models have a limited ability to take local syntactic information, they fail to use high-level and large-scale structures of documents. This information is valuable for text understanding since it contains the author’s strategy to express information, in building an effective representation and forming appropriate output modes. We propose a neural net-based model, Zooming Network, capable of representing and leveraging text structure of long document and developing its own analyzing rhythm to extract critical information. Generally, ZN consists of an encoding neural net that can build a hierarchical representation of a document, and an interpreting neural model that can read the information at multi-levels and issuing labeling actions through a policy-net. Our model is trained with a hybrid paradigm of supervised learning (distinguishing right and wrong decision) and reinforcement learning (determining the goodness among multiple right paths). We applied the proposed model to long text sequence labeling tasks, with performance exceeding baseline model (biLSTM-crf) by 10 F1-measure. … **Multiscale Artificial Neural Network (MsANN)**

Multigrid modeling algorithms are a technique used to accelerate relaxation models running on a hierarchy of similar graphlike structures. We introduce and demonstrate a new method for training neural networks which uses multilevel methods. Using an objective function derived from a graph-distance metric, we perform orthogonally-constrained optimization to find optimal prolongation and restriction maps between graphs. We compare and contrast several methods for performing this numerical optimization, and additionally present some new theoretical results on upper bounds of this type of objective function. Once calculated, these optimal maps between graphs form the core of Multiscale Artificial Neural Network (MsANN) training, a new procedure we present which simultaneously trains a hierarchy of neural network models of varying spatial resolution. Parameter information is passed between members of this hierarchy according to standard coarsening and refinement schedules from the multiscale modelling literature. In our machine learning experiments, these models are able to learn faster than default training, achieving a comparable level of error in an order of magnitude fewer training examples. … **Fusion Graph Convolutional Network**

Semi-supervised node classification involves learning to classify unlabelled nodes given a partially labeled graph. In transductive learning, all unlabelled nodes to be classified are observed during training and in inductive learning, predictions are to be made for nodes not seen at training. In this paper, we focus on both these settings for node classification in attributed graphs, i.e., graphs in which nodes have additional features. State-of-the-art models for node classification on such attributed graphs use differentiable recursive functions. These differentiable recursive functions enable aggregation and filtering of neighborhood information from multiple hops (depths). Despite being powerful, these variants are limited in their ability to combine information from different hops efficiently. In this work, we analyze this limitation of recursive graph functions in terms of their representation capacity to effectively capture multi-hop neighborhood information. Further, we provide a simple fusion component which is mathematically motivated to address this limitation and improve the existing models to explicitly learn the importance of information from different hops. This proposed mechanism is shown to improve over existing methods across 8 popular datasets from different domains. Specifically, our model improves the Graph Convolutional Network (GCN) and a variant of Graph SAGE by a significant margin providing highly competitive state-of-the-art results. …

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