In this paper, we provide a new construction for studying parameterized persistence, called a canopy. We give two versions of this construction: the A-canopy, retaining all information about points on the diagonal of the persistence diagram; and the D-canopy, encoding the information of the "standard" persistence diagram. We do this by making a simple but major modification in the persistence bundle representation information: namely, rather than tracking a point in the persistence diagram, we instead track some choice of pairs of simplices that created said point. This viewpoint is a combinatorial version of tracking the chain complex information rather than just the output of persistence. We show how to construct the canopies from any filtered filtration function, proving, using the algebraic structure of filtered chain complexes, that different choices of pairs result in homeomorphic structures. Finally, we showcase the power of our approach by using canopies to define vines even in the presence of points with multiplicity; to discuss monodromy; and to obtain some immediate results linking non-trivial monodromy in the persistent homology transform with the existence of non-Hausdorff points in the canopy.

Summary (1) Rationale Quantifying and predicting plant morphology is central to understanding development and evolution, yet many plant forms lack homologous features required for traditional morphometrics. We apply the Euler Characteristic Transform (ECT), an injective descriptor from topological data analysis, to encode 2D plant shapes. The ECT converts contours into image-like representations that preserve shape information while enabling deep learning. (2) Methods We computed ECTs for large datasets of leaf and pavement cell shapes and used convolutional neural networks (CNNs) for classification. We also trained CNNs to approximate the inverse mapping, predicting leaf shape masks from radial ECTs. (3) Key results ECT-based models achieved high classification accuracy, surpassing previous approaches on millions of herbarium-derived leaves. Notably, grapevine leaf venation was predicted from blade geometry alone, demonstrating that vascular structure is encoded in the outline. (4) Main conclusion The ECT provides a compact, information-preserving representation of biological shape that integrates naturally with deep learning. It enables both accurate classification and predictive reconstruction, revealing latent morphological information and offering new opportunities to study plant form across scales.

The leaf blade and vasculature develop together within a shared morphological space. Despite shared molecular patterning pathways, it is unknown if developmental and evolutionary variation affect these tissues separately or together in a coordinated way. Grapevine leaves have a morphometric history and abundant data measuring the shape of the blade and vasculature together. Using a combination of topological data analysis and deep learning, we perform reciprocal semantic segmentation of leaf blade and vasculature. Each tissue contains sufficient information to predict the other. We hypothesize that this is due to a one-to-one relationship between blade and vein. Using thin plate splines to swap and warp different combinations of blade and vein shapes, we show that a set of leaves with a many-to-one relationship of blade and vein are distinguishable from true leaves. We also swap blade and vein across the developmental series and between species and show that only reversing the developmental series disrupts the relationship between blade and vasculature. We end by discussing the evolutionary and developmental implications that there is a unique, one-to-one mapping between blade and vein that allows each to be predicted from the other. ### Competing Interest Statement This study was conducted in collaboration with E & J Gallo Winery, which provided access to commercial vineyards throughout California and from which data were collected. Peter Cousins, an author of this study, is an employee of E & J Gallo Winery. Algerian Ministry of Higher Education and Scientific Research, D01N01UN070120220001 National Science Foundation Plant Genome Research Program, IOS-2310355, IOS-2310356, IOS-2310357 Canada Research Chairs Program NSF Postdoctoral Research Fellowship, 2305703

Directed graphs arise in many applications where computing persistent homology helps to encode the shape and structure of the input information. However, there are only a few ways to turn the directed graph information into an undirected simplicial complex filtration required by the standard persistent homology framework. In this paper, we present a new filtration constructed from a directed graph, called the walk-length filtration. This filtration mirrors the behavior of small walks visiting certain collections of vertices in the directed graph. We show that, while the persistence is not stable under the usual L_∞-style network distance, a generalized L_1-style distance is, indeed, stable. We further provide an algorithm for its computation, and investigate the behavior of this filtration in examples, including cycle networks and synthetic hippocampal networks with a focus on comparison to the often used Dowker filtration.

The interleaving distance is arguably the most widely used metric in topological data analysis (TDA) due to its applicability to a wide array of inputs of interest, such as (multiparameter) persistence modules, Reeb graphs, merge trees, and zigzag modules. However, computation of the interleaving distance in the vast majority of this settings is known to be NP-hard, limiting its use in practical settings. Inspired by the work of Chambers et al. on the interleaving distance for mapper graphs, we solve a more general problem bounding the interleaving distance between generalized persistence modules on concrete categories via a loss function. This loss function measures how far an assignment, which can be thought of as an interleaving that might not commute, is from defining a true interleaving. We give settings for which the loss can be computed in polynomial time, including for certain assumptions on k-parameter persistence modules.

Neural signals are high-dimensional, noisy, and dynamic, making it challenging to extract interpretable features linked to behavior or disease. We introduce Neurospectrum, a framework that encodes neural activity as latent trajectories shaped by spatial and temporal structure. At each timepoint, signals are represented on a graph capturing spatial relationships, with a learnable attention mechanism highlighting important regions. These are embedded using graph wavelets and passed through a manifold-regularized autoencoder that preserves temporal geometry. The resulting latent trajectory is summarized using a principled set of descriptors - including curvature, path signatures, persistent homology, and recurrent networks -that capture multiscale geometric, topological, and dynamical features. These features drive downstream prediction in a modular, interpretable, and end-to-end trainable framework. We evaluate Neurospectrum on simulated and experimental datasets. It tracks phase synchronization in Kuramoto simulations, reconstructs visual stimuli from calcium imaging, and identifies biomarkers of obsessive-compulsive disorder in fMRI. Across tasks, Neurospectrum uncovers meaningful neural dynamics and outperforms traditional analysis methods.

The kinetic data structure (KDS) framework is a powerful tool for maintaining various geometric configurations of continuously moving objects. In this work, we introduce the kinetic hourglass, a novel KDS implementation designed to compute the bottleneck distance for geometric matching problems. We detail the events and updates required for handling general graphs, accompanied by a complexity analysis. Furthermore, we demonstrate the utility of the kinetic hourglass by applying it to compute the bottleneck distance between two persistent homology transforms (PHTs) derived from shapes in ℝ^2, which are topological summaries obtained by computing persistent homology from every direction in 𝕊^1.

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.

Data consisting of a graph with a function mapping into 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.

Mapper graphs are a widely used tool in topological data analysis and visualization. They can be viewed as discrete approximations of Reeb graphs, offering insight into the shape and connectivity of complex data. Given a high-dimensional point cloud 𝕏 equipped with a function f: 𝕏→ℝ, a mapper graph provides a summary of the topological structure of 𝕏 induced by f, where each node represents a local neighborhood, and edges connect nodes whose corresponding neighborhoods overlap. Our focus is the interleaving distance for mapper graphs, arising from a discretization of the version for Reeb graphs, which is NP-hard to compute. This distance quantifies the similarity between two mapper graphs by measuring the extent to which they must be “stretched" to become comparable. Recent work introduced a loss function that provides an upper bound on the interleaving distance for mapper graphs, which evaluates how far a given assignment is from being a true interleaving. Finding the loss is computationally tractable, offering a practical way to estimate the distance. In this paper, we employ a categorical formulation of mapper graphs and develop the first framework for computing the associated loss function. Since the quality of the bound depends on the chosen assignment, we optimize this loss function by formulating the problem of finding the best assignment as an integer linear programming problem. To evaluate the effectiveness of our optimization, we apply it to small mapper graphs where the interleaving distance is known, demonstrating that the optimized upper bound successfully matches the interleaving distance in these cases. Additionally, we conduct an experiment on the MPEG-7 dataset, computing the pairwise optimal loss on a collection of mapper graphs derived from images and leveraging the distance bound for image classification.

The Euler Characteristic Transform (ECT) is a robust method for shape classification. It takes an embedded shape and, for each direction, computes a piecewise constant function representing the Euler Characteristic of the shape's sublevel sets, which are defined by the height function in that direction. It has applications in TDA inverse problems, such as shape reconstruction, and is also employed with machine learning methodologies. In this paper, we define a distance between the ECTs of two distinct geometric embeddings of the same abstract simplicial complex and provide an upper bound for this distance. The Super Lifted Euler Characteristic Transform (SELECT), a related construction, extends the ECT to scalar fields defined on shapes. We establish a similar distance bound for SELECT, specifically when applied to fields defined on embedded simplicial complexes.

Changes in the parameters of dynamical systems can cause the state of the system to shift between different qualitative regimes. These shifts, known as bifurcations, are critical to study as they can indicate when the system is about to undergo harmful changes in its behavior. In stochastic dynamical systems, there is particular interest in P-type (phenomenological) bifurcations, which can include transitions from a monostable state to multi-stable states, the appearance of stochastic limit cycles and other features in the probability density function (PDF) of the system’s state. Current practices are limited to systems with small state spaces, cannot detect all possible behaviors of the PDFs and mandate human intervention for visually identifying the change in the PDF. In contrast, this study presents a new approach based on Topological Data Analysis that uses superlevel persistence to mathematically quantify P-type bifurcations in stochastic systems through a “homological bifurcation plot”—which shows the changing ranks of 0th and 1st homology groups, through Betti vectors. Using these plots, we demonstrate the successful detection of P-bifurcations on the stochastic Duffing, Raleigh-Vander Pol and Quintic Oscillators given their analytical PDFs, and elaborate on how to generate an estimated homological bifurcation plot given a kernel density estimate (KDE) of these systems by employing a tool for finding topological consistency between PDFs and KDEs.

AbstractPersian walnuts (Juglans regia L.) are the second most produced and consumed tree nut, with over 2.6 million metric tons produced in the 2022–2023 harvest cycle alone. The United States is the second largest producer, accounting for 25% of the total global supply. Nonetheless, producers face an ever‐growing demand in a more uncertain climate landscape, which requires effective and efficient walnut selection and breeding of new cultivars with increased kernel content and easy‐to‐open shells. Past and current efforts select for these traits using hand‐held calipers and eye‐based evaluations. Yet there is plenty of morphology that meets the eye but goes unmeasured, such as the volume of inner air or the convexity of the kernel. Here, we study the shape of walnut fruits based on X‐ray computed tomography three‐dimensional reconstructions. We compute 49 different morphological phenotypes for 1264 individual nuts comprising 149 accessions. These phenotypes are complemented by traits of breeding interest such as ease of kernel removal and kernel‐to‐nut weight ratio. Through allometric relationships, relative growth of one tissue to another, we identify possible biophysical constraints at play during development. We explore multiple correlations between all morphological and commercial traits and identify which morphological traits can explain the most variability of commercial traits. We show that using only volume‐ and thickness‐based traits, especially inner air content, we can successfully encode several of the commercial traits.

Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric graphs via merge trees. In order to preserve as much useful information as possible from the original data, we introduce a way of rotating the sublevel set to obtain the merge trees via the idea of the directional transform. We represent the merge trees using a surjective multi-labeling scheme and then compute the distance between two representative matrices. We show some theoretically desirable qualities and present two methods of computation: approximation via sampling and exact distance using a kinetic data structure, both in polynomial time. We illustrate its utility by implementing it on two data sets.

We explore a novel application of zero-dimensional persistent homology from Topological Data Analysis (TDA) for bracketing zero-crossings of both one-dimensional continuous functions, and uniformly sampled time series. We present an algorithm and show its robustness in the presence of noise for a range of sampling frequencies. In comparison to state-of-the-art software-based methods for finding zeros of a time series, our method generally converges faster, provides higher accuracy, and is capable of finding all the roots in a given interval instead of converging only to one of them. We also present and compare options for automatically setting the persistence threshold parameter that influences the accurate bracketing of the roots.

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.

Premise:The selection of Arabidopsis as a model organism played a pivotal role in advancing genomic science. The competing frameworks to select an agricultural- or ecological-based model species were rejected, in favor of building knowledge in a species that would facilitate genome-enabled research. Methods:Here, we examine the ability of models based on Arabidopsis gene expression data to predict tissue identity in other flowering plants. Comparing different machine learning algorithms, models trained and tested on Arabidopsis data achieved near perfect precision and recall values, whereas when tissue identity is predicted across the flowering plants using models trained on Arabidopsis data, precision values range from 0.69 to 0.74 and recall from 0.54 to 0.64. Results:The identity of belowground tissue can be predicted more accurately than other tissue types, and the ability to predict tissue identity is not correlated with phylogenetic distance from Arabidopsis. k-nearest neighbors is the most successful algorithm, suggesting that gene expression signatures, rather than marker genes, are more valuable to create models for tissue and cell type prediction in plants. Discussion:Our data-driven results highlight that the assertion that knowledge from Arabidopsis is translatable to other plants is not always true. Considering the current landscape of abundant sequencing data, we should reevaluate the scientific emphasis on Arabidopsis and prioritize plant diversity.

One of the most important problems arising in time series analysis is that of classifying the states of a dynamical system. That is, given a collection of time series, is it possible to perform two -state classification (chaotic versus periodic) of the underlying system? For this task, we turn to the field of topological data analysis, which encodes information about the shape and structure of data. In this paper, we investigate a more recent method for encoding the structure of the attractor as a weighted graph, known as the ordinal partition network, representing information about when the dynamical system has passed between certain regions of state space. We provide methods to incorporate the weighting information and show that this framework provides more resilience to noise or perturbations in the system as well as improving the accuracy of dynamic state identification.