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We present a method for computing an approximate Rieman-nian barycenter of a collection of points lying on a Riemannian mani-fold. Our approach relies on the use of theoretically proven under- and over-approximations of the Riemannian distance function. We compare it to Riemannian steepest descent on the exact objective function of the Riemannian barycenter and to an approach that approximates the Rie-mannian logarithm using lifting maps. Experiments are conducted on the Stiefel manifold. 

This work introduces TopoBench, an open-source library designed to standardize benchmarking and accelerate research in topological deep learning (TDL). TopoBench decomposes TDL into a sequence of independent modules for data generation, loading, transforming and processing, as well as model training, optimization and evaluation. This modular organization provides flexibility for modifications and facilitates the adaptation and optimization of various TDL pipelines. A key feature of TopoBench is its support for transformations and lifting across topological domains. Mapping the topology and features of a graph to higher-order topological domains, such as simplicial and cell complexes, enables richer data representations and more fine-grained analyses. The applicability of TopoBench is demonstrated by benchmarking several TDL architectures across diverse tasks and datasets. 

Image datasets in specialized fields of science, such as biomedicine, are typically smaller than traditional machine learning datasets. As such, they present a problem for training many models. To address this challenge, researchers often attempt to incorporate priors, i.e., external knowledge, to help the learning procedure. Geometric priors, for example, offer to restrict the learning process to the manifold to which the data belong. However, learning on manifolds is sometimes computationally intensive to the point of being prohibitive. Here, we ask a provocative question: is machine learning on manifolds really more accurate than its linear counterpart to the extent that it is worth sacrificing significant speedup in computation? We answer this question through an extensive theoretical and experimental study of one of the most common learning methods for manifold-valued data: geodesic regression. 

Women are at higher risk of Alzheimer’s and other neurological diseases after menopause, and yet research con- necting female brain health to sex hormone fluctuations is limited. We seek to investigate this connection by develop- ing tools that quantify 3D shape changes that occur in the brain during sex hormone fluctuations. Geodesic regression on the space of 3D discrete surfaces offers a principled way to characterize the evolution of a brain’s shape. However, in its current form, this approach is too computationally expensive for practical use. In this paper, we pro- pose approximation schemes that accelerate geodesic regression on shape spaces of 3D discrete surfaces. We also provide rules of thumb for when each approximation can be used. We test our approach on synthetic data to quantify the speed-accuracy trade-off of these approximations and show that practitioners can expect very significant speed-up while only sacrificing little accuracy. Finally, we apply the method to real brain shape data and produce the first characterization of how the female hippocampus changes shape during the menstrual cycle as a function of progesterone: a characterization made (practically) possible by our approximation schemes. Our work paves the way for comprehensive, practical shape analyses in the fields of bio-medicine and computer vision. Our implementation is publicly avail- able on GitHub. 

Septins self-assemble into polymers that bind and deform membranes in vitro and regulate diverse cell behaviors in vivo. How their in vitro properties relate to their in vivo functions is under active investigation. Here, we uncover requirements for septins in detachment and motility of border cell clusters in the Drosophila ovary. Septins and myosin colocalize dynamically at the cluster periphery and share phenotypes but, surprisingly, do not impact each other. Instead, Rho independently regulates myosin activity and septin localization. Active Rho recruits septins to membranes, whereas inactive Rho sequesters septins in the cytoplasm. Mathematical analyses identify how manipulating septin expression levels alters cluster surface texture and shape. This study shows that the level of septin expression differentially regulates surface properties at different scales. This work suggests that downstream of Rho, septins tune surface deformability while myosin controls contractility, the combination of which governs cluster shape and movement.