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Abstract We define a metric—the network Gromov–Wasserstein distance—on weighted, directed networks that is sensitive to the presence of outliers. In addition to proving its theoretical properties, we supply network invariants based on optimal transport that approximate this distance by means of lower bounds. We test these methods on a range of simulated network datasets and on a dataset of real-world global bilateral migration. For our simulations, we define a network generative model based on the stochastic block model. This may be of independent interest for benchmarking purposes. 

Abstract Proliferation of high‐resolution imaging data in recent years has led to substantial improvements in the two popular approaches for analyzing shapes of data objects based on landmarks and/or continuous curves. We provide an expository account of elastic shape analysis of parametric planar curves representing shapes of two‐dimensional (2D) objects by discussing its differences, and its commonalities, to the landmark‐based approach. Particular attention is accorded to the role of reparameterization of a curve, which in addition to rotation, scaling and translation, represents an important shape‐preserving transformation of a curve. The transition to the curve‐based approach moves the mathematical setting of shape analysis from finite‐dimensional non‐Euclidean spaces to infinite‐dimensional ones. We discuss some of the challenges associated with the infinite‐dimensionality of the shape space, and illustrate the use of geometry‐based methods in the computation of intrinsic statistical summaries and in the definition of statistical models on a 2D imaging dataset consisting of mouse vertebrae. We conclude with an overview of the current state‐of‐the‐art in the field. This article is categorized under: Image and Spatial Data < Data: Types and StructureComputational Mathematics < Applications of Computational Statistics 

Summary We propose a curve-based Riemannian geometric approach for general shape-based statistical analyses of tumours obtained from radiologic images. A key component of the framework is a suitable metric that enables comparisons of tumour shapes, provides tools for computing descriptive statistics and implementing principal component analysis on the space of tumour shapes and allows for a rich class of continuous deformations of a tumour shape. The utility of the framework is illustrated through specific statistical tasks on a data set of radiologic images of patients diagnosed with glioblastoma multiforme, a malignant brain tumour with poor prognosis. In particular, our analysis discovers two patient clusters with very different survival, subtype and genomic characteristics. Furthermore, it is demonstrated that adding tumour shape information to survival models containing clinical and genomic variables results in a significant increase in predictive power.