We propose a two-sample test for the isometry of two manifolds by combining ideas from spectral analysis and topological data analysis. More specifically, the proposed test statistic is the bottleneck distance between the persistence diagrams generated by the estimated heat kernel signatures (HKS) of the two manifolds. Under certain assumptions, the asymptotic normality of this test statistic can be derived, and a bootstrap approach is used to conduct the test. We also show simulation results to validate our testing model under isometric and non-isometric cases.
Jillian Eddy
(Machine) Learning the Curvature of Polyhedra
Discrete notions of Curvature in networks, graphs and simplicial complexes have been studied due to applications in data science and network analysis. In short, the curvature of a graph gives important information such how connected it is. By applying discrete curvature metrics to (the graphs of) polytopes, we can characterize their geometry and control desirable characteristics like their diameter. Through the use of online graph databases and randomly generated polytopes, we produced thousands of graphs and data corresponding to their graph-theoretical features. We developed various classification models to predict if a graph has everywhere positive curvature or not, and these models have proved incredibly accurate. Saliency analysis of these models allows us to identify and prioritize the graph features that influence curvature predictions. We hope to use findings from our machine learning to inform mathematical theorems on polytopes.
Brian Knight
Dynamics of Pole Cell Formation in Drosophila
Primordial germ cells (PGCs) are the first to form during Drosophila embryogenesis. Previous studies of PGC formation [1] show an interesting budding behavior along the embryo’s membrane as nuclei reach the embryonic cortex during the end of the 9th nuclear cycle. Notably, PGCs are initially ‘incomplete’ cells, without fully closed cell membranes, and thus standard cell segmentation and tracking procedures cannot be readily applied. We are interested in quantifying these dynamics and measuring related quantities (such as cell surface areas, volumes, germ plasm concentrations) as early in the formation process as possible in order to study variation at the cellular level. Such variation may influence the future function of the cell and its daughters. In order to study such dynamics we train a neural network on our 3D microscopy data using the method described in [2]. This network predicts a 3D displacement vector field (DVF) between two volumes. We then form a 2D surface mesh of the embryo for any given volume in order to validate our method and segment and track PGCs across multiple time points.
Brittney Marsters
Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones
Motivated by problems in Mixed Integer Conic Optimization, we investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set with the help of a group action. We show this is true for the cone of positive semidefinite matrices (PSD) and the second-order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, under the action of a finitely generated group.
Santiago Morales and Esteban Leiva
An Exact Method for Reliable Shortest Path Problems with Correlation
This poster presents a novel approach to solving Reliable Shortest Path Problems (RSPPs) in road networks with uncertain, correlated travel times. While traditional Least Expected Travel Time (LET) paths fail to account for the high variability and unreliability experienced in real-world scenarios, we focus on RSP objectives that prioritize reliability. Key objectives such as the α-reliable shortest path (α-RSP), maximum probability of on-time arrival path (MPOAP), and the shortest α-reliable path (S-αRP) are considered, each addressing different reliability metrics. Travel times are modeled using a joint normal distribution with non-negative correlations, which is more representative of real road networks, where correlations significantly impact path reliability. This is the first study to provide an exact solution for the S-αRP under correlated travel times. We propose an algorithm based on a recursive depth-first search procedure, known as the pulse algorithm, that solves various RSP objectives within this stochastic, correlated setting. The algorithm uses an upper bound on partial path reliability and demonstrates competitive performance across large-scale road networks, outperforming α-RSP existing state-of-the-art methods, such as dominance-based A* algorithms. The algorithm is implemented in a fully documented Julia package, providing a practical tool for routing applications under uncertainty.
Yunpeng Shi
Fast and robust alignment of images in filtered sliced Wasserstein distance
We present a fast algorithm for aligning images using optimal transport. Our method is based on the sliced Wasserstein distance and computes the 1-D Wasserstein distance between radial line projections of the input images. Using this approach, we develop an algorithm that can align two L by L images in O(L^2 log L) operations. This complexity matches the one that uses the Euclidean distance. We show that our method is robust to rotations, translations and deformations in the images.
Tait Weicht
More applications for linear intersections of conic varieties: projected multi-reference alignment
Single particle cryogenic electron microscopy (cryo-EM) is a ground breaking technique for determining the structure of a molecule. Cryo-EM inspired a simpler model for structure determination, multi-reference alignment (MRA), which captures some of the difficulties in recovering a signal. We modify the MRA model to include a step modeling tomographic projection. Naively, recovering the signal in our model would require a tensor decomposition method in a regime where no known algorithm can succeed. However, we find a new and interesting strategy based on an existing algorithm for computing linear intersections with Segre varieties that in experiments recovers the signal parameters.