A multivariate Hawkes process enables self- and cross-excitations through a triggering matrix that behaves like an asymmetrical covariance structure, characterizing pairwise interactions between the event types. Full-rank estimation of all interactions is often infeasible in empirical settings. Models that specialize on a spatiotemporal application alleviate this obstacle by exploiting spatial locality, allowing the dyadic relationships between events to depend only on separation in time and relative distances in real Euclidean space. Here we generalize this framework to any multivariate Hawkes process, and harness it as a vessel for embedding arbitrary event types in a hidden metric space. Specifically, we propose a Hidden Hawkes Geometry (HHG) model to uncover the hidden geometry between event excitations in a multivariate point process. The low dimensionality of the embedding regularizes the structure of the inferred interactions. We develop a number of estimators and validate the model by conducting several experiments. In particular, we investigate regional infectivity dynamics of COVID-19 in an early South Korean record and recent Los Angeles confirmed cases. By additionally performing synthetic experiments on short records as well as explorations into options markets and the Ebola epidemic, we demonstrate that learning the embedding alongside a point process uncovers salient interactions in a broad range of applications.

Surface roughness and texture are critical to the functional performance of engineering components. The ability to analyze roughness and texture effectively and efficiently is much needed to ensure surface quality in many surface generation processes, such as machining, surface mechanical treatment, etc. Discrete Wavelet Transform (DWT) and Discrete Cosine Transform (DCT) are two commonly used signal decomposition tools for surface roughness and texture analysis. Both methods require selecting a threshold to decompose a given surface into its three main components: form, waviness, and roughness. However, although DWT and DCT are part of the ISO surface finish standards, there exists no systematic guidance on how to compute these thresholds, and they are often manually selected on case by case basis. This makes utilizing these methods for studying surfaces dependent on the user's judgment and limits their automation potential. Therefore, we present two automatic threshold selection algorithms based on information theory and signal energy. We use machine learning to validate the success of our algorithms both using simulated surfaces as well as digital microscopy images of machined surfaces. Specifically, we generate feature vectors for each surface area or profile and apply supervised classification. Comparing our results with the heuristic threshold selection approach shows good agreement with mean accuracies as high as 95\%. We also compare our results with Gaussian filtering (GF) and show that while GF results for areas can yield slightly higher accuracies, our results outperform GF for surface profiles. We further show that our automatic threshold selection has significant advantages in terms of computational time as evidenced by decreasing the number of mode computations by an order of magnitude compared to the heuristic thresholding for DCT.

A topic model is often formulated as a generative model that explains how each word of a document is generated given a set of topics and document-specific topic proportions. It is focused on capturing the word co-occurrences in a document and hence often suffers from poor performance in analyzing short documents. In addition, its parameter estimation often relies on approximate posterior inference that is either not scalable or suffers from large approximation error. This paper introduces a new topic-modeling framework where each document is viewed as a set of word embedding vectors and each topic is modeled as an embedding vector in the same embedding space. Embedding the words and topics in the same vector space, we define a method to measure the semantic difference between the embedding vectors of the words of a document and these of the topics, and optimize the topic embeddings to minimize the expected difference over all documents. Experiments on text analysis demonstrate that the proposed method, which is amenable to mini-batch stochastic gradient descent based optimization and hence scalable to big corpora, provides competitive performance in discovering more coherent and diverse topics and extracting better document representations.

We present a framework for embedding graph structured data into a vector space, taking into account node features and topology of a graph into the optimal transport (OT) problem. Then we propose a novel distance between two graphs, named linearFGW, defined as the Euclidean distance between their embeddings. The advantages of the proposed distance are twofold: 1) it can take into account node feature and structure of graphs for measuring the similarity between graphs in a kernel-based framework, 2) it can be much faster for computing kernel matrix than pairwise OT-based distances, particularly fused Gromov-Wasserstein, making it possible to deal with large-scale data sets. After discussing theoretical properties of linearFGW, we demonstrate experimental results on classification and clustering tasks, showing the effectiveness of the proposed linearFGW.

However, evaluating the OT incurs a high computational complexity in Optimal transport (OT) is a popular measure general (Peyré and Cuturi, 2019) which leads to several to compare probability distributions. However, proposals in the recent literature to address this drawback OT suffers a few drawbacks such as (i) of OT, e.g., approximate using entropic regularization a high complexity for computation, (ii) indefiniteness (Cuturi, 2013), or exploit geometric structure which limits its applicability to of supports (Rabin et al., 2011; Le et al., 2019; Le and kernel machines. In this work, we consider Nguyen, 2021). Among them, tree-Wasserstein (Evans probability measures supported on a graph and Matsen, 2012; Le et al., 2019) (TW) leverages the metric space and propose a novel Sobolev tree structure over supports to obtain a closed-form transport metric. We show that the Sobolev for fast computation. However, the requirement about transport metric yields a closed-form formula tree structure for supports may be restricted in applications.

Determining whether saddle points exist or are approximable for nonconvex-nonconcave problems is usually intractable. We take a step towards understanding certain nonconvex-nonconcave minimax problems that do remain tractable. Specifically, we study minimax problems cast in geodesic metric spaces, which provide a vast generalization of the usual convex-concave saddle point problems. The first main result of the paper is a geodesic metric space version of Sion's minimax theorem; we believe our proof is novel and transparent, as it relies on Helly's theorem only. In our second main result, we specialize to geodesically complete Riemannian manifolds: we devise and analyze the complexity of first-order methods for smooth minimax problems.

Randomized smoothing is considered to be the state-of-the-art provable defense against adversarial perturbations. However, it heavily exploits the fact that classifiers map input objects to class probabilities and do not focus on the ones that learn a metric space in which classification is performed by computing distances to embeddings of classes prototypes. In this work, we extend randomized smoothing to few-shot learning models that map inputs to normalized embeddings. We provide analysis of Lipschitz continuity of such models and derive robustness certificate against $\ell_2$-bounded perturbations that may be useful in few-shot learning scenarios. Our theoretical results are confirmed by experiments on different datasets.

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Frechet mean. In this work, we equip a set of graphs with the pseudometric defined by the norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Frechet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric.

Understanding the operation of biological and artificial networks remains a difficult and important challenge. To identify general principles, researchers are increasingly interested in surveying large collections of networks that are trained on, or biologically adapted to, similar tasks. A standardized set of analysis tools is now needed to identify how network-level covariates -- such as architecture, anatomical brain region, and model organism -- impact neural representations (hidden layer activations). Here, we provide a rigorous foundation for these analyses by defining a broad family of metric spaces that quantify representational dissimilarity. Using this framework we modify existing representational similarity measures based on canonical correlation analysis to satisfy the triangle inequality, formulate a novel metric that respects the inductive biases in convolutional layers, and identify approximate Euclidean embeddings that enable network representations to be incorporated into essentially any off-the-shelf machine learning method. We demonstrate these methods on large-scale datasets from biology (Allen Institute Brain Observatory) and deep learning (NAS-Bench-101). In doing so, we identify relationships between neural representations that are interpretable in terms of anatomical features and model performance.

Modeling heterogeneity by extraction and exploitation of high-order information from heterogeneous information networks (HINs) has been attracting immense research attention in recent times. Such heterogeneous network embedding (HNE) methods effectively harness the heterogeneity of small-scale HINs. However, in the real world, the size of HINs grow exponentially with the continuous introduction of new nodes and different types of links, making it a billion-scale network. Learning node embeddings on such HINs creates a performance bottleneck for existing HNE methods that are commonly centralized, i.e., complete data and the model are both on a single machine. To address large-scale HNE tasks with strong efficiency and effectiveness guarantee, we present \textit{Decentralized Embedding Framework for Heterogeneous Information Network} (DeHIN) in this paper. In DeHIN, we generate a distributed parallel pipeline that utilizes hypergraphs in order to infuse parallelization into the HNE task. DeHIN presents a context preserving partition mechanism that innovatively formulates a large HIN as a hypergraph, whose hyperedges connect semantically similar nodes. Our framework then adopts a decentralized strategy to efficiently partition HINs by adopting a tree-like pipeline. Then, each resulting subnetwork is assigned to a distributed worker, which employs the deep information maximization theorem to locally learn node embeddings from the partition it receives. We further devise a novel embedding alignment scheme to precisely project independently learned node embeddings from all subnetworks onto a common vector space, thus allowing for downstream tasks like link prediction and node classification.