This study explores the dynamic relationship between online discourse, as observed in tweets, and physical hate crimes, focusing on marginalized groups. Leveraging natural language processing techniques, including keyword extraction and topic modeling, we analyze the evolution of online discourse after events affecting these groups. Examining sentiment and polarizing tweets, we establish correlations with hate crimes in Black and LGBTQ+ communities. Using a knowledge graph, we connect tweets, users, topics, and hate crimes, enabling network analyses. Our findings reveal divergent patterns in the evolution of user communities for Black and LGBTQ+ groups, with notable differences in sentiment among influential users. This analysis sheds light on distinctive online discourse patterns and emphasizes the need to monitor hate speech to prevent hate crimes, especially following significant events impacting marginalized communities.

Graph neural networks (GNNs) achieve remarkable performance in graph machine learning tasks but can be hard to train on large-graph data, where their learning dynamics are not well understood. We investigate the training dynamics of large-graph GNNs using graph neural tangent kernels (GNTKs) and graphons. In the limit of large width, optimization of an overparametrized NN is equivalent to kernel regression on the NTK. Here, we investigate how the GNTK evolves as another independent dimension is varied: the graph size. We use graphons to define limit objects---graphon NNs for GNNs, and graphon NTKs for GNTKs---, and prove that, on a sequence of graphs, the GNTKs converge to the graphon NTK. We further prove that the spectrum of the GNTK, which is related to the problem's learning directions, converges to the spectrum of the GNTK. This implies that in the large-graph limit, the GNTK fitted on a graph of moderate size can be used to solve the same task on the large graph, and to infer the learning dynamics of the large-graph GNN. These results are verified empirically on node regression and classification tasks.

While variational autoencoders have been successful in several tasks, the use of conventional priors are limited in their ability to encode the underlying structure of input data. We introduce an Encoded Prior Sliced Wasserstein AutoEncoder wherein an additional prior-encoder network learns a geometry and topology preserving embedding of any data manifold, thus improving the structure of latent space. The autoencoder and prior-encoder networks are iteratively trained using the Sliced Wasserstein distance, which facilitates the learning of nonstandard complex priors. We then introduce a graph-based algorithm to explore the learned manifold by traversing latent space through network-geodesics that lie along the manifold and hence are more realistic compared to conventional Euclidean interpolation. Specifically, we identify network-geodesics by maximizing the density of samples along the path while minimizing total energy. We use the 3D-spiral data to show that the prior encodes the geometry underlying the data unlike conventional autoencoders, and to demonstrate the exploration of the embedded data manifold through the network algorithm. We apply our framework to artificial as well as image datasets to demonstrate the advantages of learning improved latent structure, outlier generation, and geodesic interpolation.

In this work, we study the dynamical properties of a machine learning technique called reservoir computing in order to gain insight into how representations of chaotic signals are encoded through learning. We train the reservoir on individual chaotic Lorenz signals. The Lorenz system is characterized by a set of equations and known to have three fixed points, all of which are unstable in the chaotic regime of the strange attractor. Exploration of the fixed points of the reservoir whose outputs are trained allows us to understand whether inherent Lorenz dynamics are transposed onto reservoir dynamics during learning. We do so by using a novel fixed point finding technique called directional fibers. Directional fibers are mathematical objects that systematically locate fixed points in a high dimensional space, and are found to be competitive and complementary with other traditional approaches. We find that the reservoir, after training of output weights, contains a higher dimensional projection of the Lorenz fixed points with matching stability, even though the training data did not include the fixed points. This tells us that the reservoir does indeed learn dynamical properties of the Lorenz attractor. We also find that the directional fiber also identifies additional fixed points in the reservoir space outside the projected Lorenz attractor region; these amplify perturbations during prediction and play a role in failure of long-term time series prediction.