Tree-like structures appear in many areas of science, and their shapes can help understand the underlying processes they drive or that give rise to them. By thinking of these structures as geometric graphs in $\mathbb{R}^3$, we gain access to tools from computational geometry and topology to study them. In this paper, we adopt the theory of quadratic forms to measure the directional spread of geometric graphs, and we introduce the hexplot model -- equipped with a metric derived from the Fisher metric on the standard triangle -- to visualize, measure, and collect statistics.

Skeletonization of tubular structures from 3D imaging is essential for tasks such as morphometric analysis, transport or flow simulation, and procedural planning in domains including vascular networks, plant root systems, and neural connectomes. However, automatic skeleton extraction often introduces topological artifacts, such as erroneous connections between nearby branches and fragmented centerlines caused by noise or missing data. Correcting these artifacts manually can be time-consuming and error-prone, especially when precise interaction is required. We present a semi-automatic correction method that reconstructs a plausible centerline connection from minimal user input. Given a user-selected source and target point, our method traces a path by combining (i) component-wise minimum spanning trees for stable local propagation and (ii) a filtered 3D Delaunay edge graph for bridging gaps and handling ambiguous junctions. Candidate steps are ranked using a score that accounts for direction continuity, spatial proximity, component consistency, and target-directed progress. The output is an ordered polyline (or edge sequence) that can be used as a suggested correction and integrated into downstream skeleton post-processing workflows. We implement the system in C++ with an interactive viewer based on Libigl and demonstrate representative qualitative results on brain vessel datasets, including correction of typical "crossing" and "dotted" artifacts. While our current validation is qualitative, the method is lightweight and serves as a practical building block toward more comprehensive interactive correction pipelines in biomedical imaging and related domains.

We study efficient algorithms for one-dimensional fixed-cardinality minimum Riesz $s$-energy subset selection on ordered real-line point sets and propose and test a polynomial-time exact s-t cut-based algorithm for this problem. Given $x_1<\cdots<x_n$, an exponent $s>0$, and a cardinality $k$, the task is to choose $1\leq i_1<\cdots<i_k\leq n$ minimizing $E_s(i_1,\ldots,i_k)=\sum_{1\leq p<q\leq k}(x_{i_q}-x_{i_p})^{-s}$. We prove that the one-dimensional Riesz interaction satisfies a Monge inequality. When feasible subsets are encoded as increasing index vectors, this property implies submodularity on a finite distributive lattice and yields polynomial-time solvability by submodular minimization over such lattices. The structural reduction holds for every real $s>0$. We also derive an explicit minimum $S$--$T$ cut formulation with $k(n-k)$ threshold variables and $O(k^2(n-k)^2)$ finite pairwise edges. The constructed graph has $N=k(n-k)$ nodes and $M=O(k^2(n-k)^2)$ arcs after an $O(k^2(n-k)^2)$ coefficient-construction step; an $O(NM)$ max-flow bound gives an $O(k^3(n-k)^3)$ cut step, while the conservative $O(N^2M)$ bound gives $O(k^4(n-k)^4)$. By an isometry argument, the same algorithm applies to $\ell_1$-staircases, including monotone two-dimensional Pareto-front and skyline approximations. The accompanying Python implementation includes verification examples and an empirical runtime benchmark; on balanced instances $n=2k$, the reference min-cut code overtakes exhaustive enumeration around $n=24$--$26$. The appendix provides examples and detailed explanations of the underlying theory.

A widely held intuition in deep learning is that stochastic gradient descent (SGD) implicitly favors flat minima and that flat minima generalize better, but standard Euclidean measures of flatness such as the trace or maximum eigenvalue of the loss Hessian are not invariant under reparametrizations that preserve the network function, which undermines the theoretical foundations of this narrative. In this study we resolve this issue by grounding flatness in the Riemannian geometry of the statistical manifold induced by the Fisher Information Matrix (FIM). We define Riemannian sharpness mathematically and prove that it is invariant under smooth, function-preserving reparametrizations, which directly addresses the critique of Dinh et al. in the paper ``Sharp minima can generalize for deep nets''.We note that this invariance is a property of the true FIM; the diagonal empirical estimator used in practice (and in all experiments below) inherits invariance only approximately, and exact invariance under arbitrary reparametrizations would require structured estimators such as K-FAC. We formalize the gradient noise of mini-batch SGD as having a covariance structure proportional to the FIM, derive the stationary distribution of the resulting stochastic differential equation, and then show that the probability mass is exponentially concentrated at Riemannian-flat minima. A PAC-Bayes generalization bound controlled explicitly by SR formally links this geometric bias to test performance. Our experiments on MNIST and CIFAR-10 confirm that SR reliably tracks generalization in ways that Euclidean sharpness does not, and that its scaling with $η/B$ matches the theoretical predictions. Together these results provide a rigorous, reparametrization-invariant account of why flat minima generalize.

An elastic space curve is a critical point of the bending energy subject to appropriate constraints. An analytic representation, equivalent to the spherical pendulum equation, leads to an 11-parameter description of the space of 3D elastic curve segments. We give a numerically stable method for recovering the 11 parameters from a given elastic curve segment. Using this, we give a fast and stable method to approximate an arbitrary space curve segment by a 3D elastica. Applications include interactive design with exact elastic curves and CAD surface rationalization for robotic hot-blade cutting.

The geometry of an object plays a vital role in modulating its interactions with the physical world. It nevertheless remains difficult to describe geometric information numerically for the purposes of statistical inference or classification tasks. Here, we introduce a new topological transform which leverages directional piecewise-linear Morse theory to quantify the geometry of an embedded object by cataloguing critical points across multiple height-functions. The output of this Morse transform records both the heights and the local topological type (peak, trough or saddle) of the critical points that characterise the underlying shape, retaining finer information than the Euler characteristic transform whilst naturally prioritising a shape's outermost regions. Crucially, this output can be further compressed into a rich but compact feature vector. We benchmark the Morse feature vector as a descriptor for ligand-based virtual screening (LBVS), which intrinsically depends on the shape of molecules. Under a common gradient-boosted tree classification pipeline, Morse descriptors achieve the highest mean AUROC when compared to other topological transform descriptors and to standard shape-based LBVS descriptors.

Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this work, we study the interleaving distance on discretization of these objects, called mapper graphs when $d=1$, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from recent work by Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial with a given assignment. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts.

The Euler characteristic transform (ECT) is a simple to define yet powerful representation of shape. The idea is to encode an embedded shape using sub-level sets of a a function defined based on a given direction, and then returning the Euler characteristics of these sublevel sets. Because the ECT has been shown to be injective on the space of embedded simplicial complexes, it has been used for applications spanning a range of disciplines, including plant morphology and protein structural analysis. In this survey article, we present a comprehensive overview of the Euler characteristic transform, highlighting the main idea on a simple leaf example, and surveying its its key concepts, theoretical foundations, and available applications.

Graphs drawn in the plane are ubiquitous, arising from data sets through a variety of methods ranging from GIS analysis to image classification to shape analysis. A fundamental problem in this type of data is comparison: given a set of such graphs, can we rank how similar they are, in such a way that we capture their geometric "shape" in the plane? In this paper we explore a method to compare two such embedded graphs, via a simplified combinatorial representation called a tail-less merge tree which encodes the structure based on a fixed direction. First, we examine the properties of a distance designed to compare merge trees called the branching distance, and show that the distance as defined in previous work fails to satisfy some of the requirements of a metric. We incorporate this into a new distance function called average branching distance to compare graphs by looking at the branching distance for merge trees defined over many directions. Despite the theoretical issues, we show that the definition is still quite useful in practice by using our open-source code to cluster data sets of embedded graphs.

Merge trees are a type of graph-based topological summary that tracks the evolution of connected components in the sublevel sets of scalar functions. They enjoy widespread applications in data analysis and scientific visualization. In this paper, we consider the problem of comparing two merge trees via the notion of interleaving distance in the metric space setting. We investigate various theoretical properties of such a metric. In particular, we show that the interleaving distance is intrinsic on the space of labeled merge trees and provide an algorithm to construct metric 1-centers for collections of labeled merge trees. We further prove that the intrinsic property of the interleaving distance also holds for the space of unlabeled merge trees. Our results are a first step toward performing statistics on graph-based topological summaries.

Reeb graphs are widely used in a range of fields for the purposes of analyzing and comparing complex spaces via a simpler combinatorial object. Further, they are closely related to extended persistence diagrams, which largely but not completely encode the information of the Reeb graph. In this paper, we investigate the effect on the persistence diagram of a particular continuous operation on Reeb graphs; namely the (truncated) smoothing operation. This construction arises in the context of the Reeb graph interleaving distance, but separately from that viewpoint provides a simplification of the Reeb graph which continuously shrinks small loops. We then use this characterization to initiate the study of inverse problems for Reeb graphs using smoothing by showing which paths in persistence diagram space (commonly known as vineyards) can be realized by a path in the space of Reeb graphs via these simple operations. This allows us to solve the inverse problem on a certain family of piecewise linear vineyards when fixing an initial Reeb graph.

In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter $τ$. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for $0 \leq τ\leq 2\varepsilon$, where $\varepsilon$ is the smoothing parameter. Then, for the restriction of $τ\in [0,\varepsilon]$, we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope $m \in [0,1]$. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every $m \in [0,1]$, which is a generalization of the original interleaving distance, which is the case $m=0$. While the resulting metrics are not stable, we show that any pair of these for $m,m' \in [0,1)$ are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs.

We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $\mathbb{X}$ equipped with a continuous function $f: \mathbb{X} \rightarrow \mathbb{R}$. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line $\mathbb{R}$. We then introduce a variant of the classic mapper graph of Singh et al.~(2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $(\mathbb{X}, f)$ when it is applied to points randomly sampled from a probability density function concentrated on $(\mathbb{X}, f)$. Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (2018), we first show that the mapper graph of $(\mathbb{X}, f)$, a constructible $\mathbb{R}$-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $(\mathbb{X},f)$ to the mapper of a super-level set of a probability density function concentrated on $(\mathbb{X}, f)$. Finally, building on the approach of Bobrowski et al.~(2017), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.

The diurnal cycle of tropical cyclones (TCs) is a daily cycle in clouds that appears in satellite images and may have implications for TC structure and intensity. The diurnal pattern can be seen in infrared (IR) satellite imagery as cyclical pulses in the cloud field that propagate radially outward from the center of nearly all Atlantic-basin TCs. These diurnal pulses, a distinguishing characteristic of the TC diurnal cycle, begin forming in the storm's inner core near sunset each day and appear as a region of cooling cloud-top temperatures. The area of cooling takes on a ring-like appearance as cloud-top warming occurs on its inside edge and the cooling moves away from the storm overnight, reaching several hundred kilometers from the circulation center by the following afternoon. The state-of-the-art TC diurnal cycle measurement has a limited ability to analyze the behavior beyond qualitative observations. We present a method for quantifying the TC diurnal cycle using one-dimensional persistent homology, a tool from Topological Data Analysis, by tracking maximum persistence and quantifying the cycle using the discrete Fourier transform. Using Geostationary Operational Environmental Satellite IR imagery data from Hurricane Felix (2007), our method is able to detect an approximate daily cycle.

There are many metrics available to compare phylogenetic trees since this is a fundamental task in computational biology. In this paper, we focus on one such metric, the $\ell^\infty$-cophenetic metric introduced by Cardona et al. This metric works by representing a phylogenetic tree with $n$ labeled leaves as a point in $\mathbb{R}^{n(n+1)/2}$ known as the cophenetic vector, then comparing the two resulting Euclidean points using the $\ell^\infty$ distance. Meanwhile, the interleaving distance is a formal categorical construction generalized from the definition of Chazal et al., originally introduced to compare persistence modules arising from the field of topological data analysis. We show that the $\ell^\infty$-cophenetic metric is an example of an interleaving distance. To do this, we define phylogenetic trees as a category of merge trees with some additional structure; namely labelings on the leaves plus a requirement that morphisms respect these labels. Then we can use the definition of a flow on this category to give an interleaving distance. Finally, we show that, because of the additional structure given by the categories defined, the map sending a labeled merge tree to the cophenetic vector is, in fact, an isometric embedding, thus proving that the $\ell^\infty$-cophenetic metric is, in fact, an interleaving distance.