Persistent Homology

Popular network models such as the mixed membership and standard stochastic block model are known to exhibit distinct geometric structure when embedded into R d using spectral methods. The resulting point cloud concentrates around a simplex in the first model, whereas it separates into clusters in the second. By adopting the formalism of generalised random dotproduct graphs in , we demonstrate that both of these models, and different mixing regimes in the case of mixed membership, may be distinguished by the persistent homology of the underlying point distribution in the case of adjacency spectral embedding. Moreover, despite non-identifiability issues, we show that the persistent homology of the support of the distribution and its super-level sets can be consistently estimated. As an application of our consistency results, we provide a topological hypothesis test for distinguishing the standard and mixed membership stochastic block models.

This paper works as a motivation to consider stronger methods in TDA (Topological Data Analysis). We discuss some of the main principles used currently in Big Data analysis. This discussion points became evident while we apply TDA to Big Data. Mathematical formal concepts might be considered, in order, to obtain better results from Data Analysis. Among these arguable principles, it may be considered "data grouping methods", and also "relation data base-graph and its connection". We discuss the topological argument, which stands as a powerful tool on Big Data analysis and TDA, from "inverse numbers" methodology, and we present a qualitative description of "Invariant Method" which we think is a strong methodology in order to obtain valuable hidden information from Big Data sets. Furthermore, we propose an algorithm to invariant method.

We compute the homology of random Čech complexes over a homogeneous Poisson process on the d-dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erdős -Rényi phase transition, where the Čech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups.

Large datasets describing geometric shapes are nowadays available in a variety of applications, including Geographical Information System. Handling such shapes require extensive computing resources. One of the most computer-intensive task is the analysis of their morphological structure. Morse Theory is a powerful theory, that is used to address the problem of describing topology, i.e., the shape features of an object and their relations. Morse theory is defined for a subset of C ∞ functions, that are called Morse functions. We have focused on morphological analysis of 2D scalar fields based on Morse Theory, in particular we have developed techniques for decomposing models according to their semantic structure. Thanks to mesh-based multi-resolution techniques, we can handle large models and dynamically extract smaller-size representations of the an object, that can be efficiently visualized and manipulated. This thesis combines a decomposition of the shape, that takes into account t...

The aim of our investigation is to use R to create a tool to estimate the relevance of random variables within a model. The algorithm will extract the geometric information from a data cloud by exploiting its topological features. It will reveal the reconstruction of the corresponding enveloping manifold.

The unknown function m : Rp 7! R describes the conditional expectation of Y given (X1, X2, . . . Xp). Suppose also that (Xi1, Xi2 . . . Xip, Yi) for i = 1, . . . , n is a sample of size n of the random vector(X1, X2 . . . Xp, Y ). If p n we will require to increase n exponentially to fit the model. One option to select the variables is to rank them according to their impact inside the model. In [1] di↵erent techniques to estimate such indicators are studied. Cited examples are: parametric and non-parametric settings, variance-based measures, moment independent measures, and graphical techniques. Using topological tools, we aim to exploit the geometrical information from clouds of random and functional data to build the afore mentioned sensitivity indicators.

Persistent Homology is a fairly new branch of Computational Topology which combines geometry and topology for an effective shape description of use in Pattern Recognition. In particular it registers through "Betti Numbers" the presence of holes and their persistence while a parameter ("filtering function") is varied. In this paper, some recent developments in this field are integrated in a k-Nearest Neighbor search algorithm suited for an automatic retrieval of melanocytic lesions. Since long, dermatologists use five morphological parameters (A = Asymmetry, B = Boundary, C = Color, D = Diameter, E = Elevation or Evolution) for assessing the malignancy of a lesion. The algorithm is based on a qualitative assessment of the segmented images by computing both 1 and 2-dimensional Persistent Betti Numbers functions related to the ABCDE parameters and to the internal texture of the lesion. The results of a feasibility test on a set of 107 melanocytic lesions are reported in the section dedicated to the numerical experiments. Massimo Ferri is full professor of Geometry at the University of Bologna. Coming from topology of low-dimensional manifolds, he grew interested in the applications of geometry and topology to robotics and to pattern recognition. His present research is in persistent topology and applications.

Node id: 1746 Category: Tch F (i) (iii) (ii) timestamp connected component G Fig. 1. TDANetVis system prototype, an interactive and web-based system with linked views designed to assist the analysis of temporal graphs by highlighting connected components' structure and evolution. (A) Colored barcode highlighting the longest connected component in the graph. (B) Timestamp markers indicating the three timestamps being depicted by (C-E) the three node-link diagrams. (F) Tooltip showing extra information. (G) Users can select groups of nodes by label or by connected component.

For medical image analysis, the test statistic of the measurements is usually constructed at every voxels in space and thresholded to determine the regions of significant signals. This thresholding produces a small patch of regions around the critical values of the test statistic. It is known that the probability of the critical values bigger than a specific threshold can be computed as the expectation of the Euler characteristic of the patch. Motivated by this topological connection, we present a new computational framework of modeling various functional measurements as topological objects. The level set associated with functional measurements can be approximated using a simplicial complex consisting of nodes and links. The existence of links basically determine the underlying topological structure of the signal. The strength of links can be modeled using an underdetermined linear model. By incorporating sparsity into the model, the links can be sparsely obtained making interpretation and visualization of the simplicial complex easier. The main contribution of this paper is showing the relationship between sparse topological structures to the sparse regression framework. We apply this novel framework in constructing a structural brain network model.

In this paper we present a novel methodology based on a topological entropy, the so-called persistent entropy, for addressing the comparison between discrete piecewise linear functions. The comparison is certified by the stability theorem for persistent entropy. The theorem is used in the implementation of a new algorithm. The algorithm transforms a discrete piecewise linear function into a filtered simplicial complex that is analyzed with persistent homology and persistent entropy. Persistent entropy is used as discriminant feature for solving the supervised classification problem of real long length noisy signals of DC electrical motors. The quality of classification is stated in terms of the area under receiver operating characteristic curve (AUC=94.52%).

Topological data analysis has been recently used to extract meaningful information from biomolecules. Here we introduce the application of persistent homology, a topological data analysis tool, for computing persistent features (loops) of the RNA folding space. The scaffold of the RNA folding space is a complex graph from which the global features are extracted by completing the graph to a simplicial complex via the notion of clique and Vietoris-Rips complexes. The resulting simplicial complexes are characterised in terms of topological invariants, such as the number of holes in any dimension, i.e. Betti numbers. Our approach discovers persistent structural features, which are the set of smallest components to which the RNA folding space can be reduced. Thanks to this discovery, which in terms of data mining can be considered as a space dimension reduction, it is possible to extract a new insight that is crucial for understanding the mechanism of the RNA folding towards the optimal secondary structure. This structure is composed by the components discovered during the reduction step of the RNA folding space and is characterized by minimum free energy.

We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets, and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method SW1PerS which stands for Sliding Windows and 1-dimensional Persistence Scoring.

This dissertation places intersection homology and local homology within the framework of persistence, which was originally developed for ordinary homology by Edelsbrunner, Letscher, and Zomorodian. The eventual goal, begun but not completed here, is to provide analytical tools for the study of embedded stratified spaces, as well as for high-dimensional and possibly noisy datasets for which the number of degrees of freedom may vary across the parameter space. Specifically, we create a theory of persistent intersection homology for a filtered stratified space and prove several structural theorems about the pair groups associated to such a filtration. We prove the correctness of a cubic algorithm which computes these pair groups in a simplicial setting. We also define a series of intersection homology elevation functions for an embedded stratified space and characterize their local maxima in dimension one. In addition, we develop a theory of persistence for a multi-scale analogue of the local homology groups of a stratified space at a point. This takes the form of a series of local homology vineyards which allow one to assess the homological structure within a one-parameter family of neighborhoods of the point. Under the assumption of dense sampling, we prove the correctness of this assessment at a variety of radius scales.

Motivated by the measurement of local homology and of functions on noisy domains, we extend the notion of persistent homology to sequences of kernels, images, and cokernels of maps induced by inclusions in a filtration of pairs of spaces. Specifically, we note that persistence in this context is well defined, we prove that the persistence diagrams are stable, and we explain how to compute them.

This dissertation places intersection homology and local homology within the framework of persistence, which was originally developed for ordinary homology by Edelsbrunner, Letscher, and Zomorodian. The eventual goal, begun but not completed here, is to provide analytical tools for the study of embedded stratified spaces, as well as for high-dimensional and possibly noisy datasets for which the number of degrees of freedom may vary across the parameter space. Specifically, we create a theory of persistent intersection homology for a filtered stratified space and prove several structural theorems about the pair groups associated to such a filtration. We prove the correctness of a cubic algorithm which computes these pair groups in a simplicial setting. We also define a series of intersection homology elevation functions for an embedded stratified space and characterize their local maxima in dimension one. In addition, we develop a theory of persistence for a multi-scale analogue of the local homology groups of a stratified space at a point. This takes the form of a series of local homology vineyards which allow one to assess the homological structure within a one-parameter family of neighborhoods of the point. Under the assumption of dense sampling, we prove the correctness of this assessment at a variety of radius scales.

In this paper we present an approach to determine the smallest possible number of neurons in a layer of a neural network in such a way that the topology of the input space can be learned sufficiently well. We introduce a general procedure based on persistent homology to investigate topological invariants of the manifold on which we suspect the data set. We specify the required dimensions precisely, assuming that there is a smooth manifold on or near which the data are located. Furthermore, we require that this space is connected and has a commutative group structure in the mathematical sense. These assumptions allow us to derive a decomposition of the underlying space whose topology is well known. We use the representatives of the k-dimensional homology groups from the persistence landscape to determine an integer dimension for this decomposition. This number is the dimension of the embedding that is capable of capturing the topology of the data manifold. We derive the theory and validate it experimentally on toy data sets.

In this paper, we use topological data analysis techniques to construct a suitable neural network classifier for the task of learning sensor signals of entire power plants according to their reference designation system. We use representations of persistence diagrams to derive necessary preprocessing steps and visualize the large amounts of data. We derive deep architectures with one-dimensional convolutional layers combined with stacked long short-term memories as residual networks suitable for processing the persistence features. We combine three separate sub-networks, obtaining as input the time series itself and a representation of the persistent homology for the zeroth and first dimension. We give a mathematical derivation for most of the used hyper-parameters. For validation, numerical experiments were performed with sensor data from four power plants of the same construction type.

By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and prove that robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.

In this work, we develop a pipeline that associates Persistence Diagrams to digital data via the most appropriate filtration for the type of data considered. Using a grid search approach, this pipeline determines optimal representation methods and parameters. The development of such a topological pipeline for Machine Learning involves two crucial steps that strongly affect its performance: firstly, digital data must be represented as an algebraic object with a proper associated filtration in order to compute its topological summary, the Persistence Diagram. Secondly, the persistence diagram must be transformed with suitable representation methods in order to be introduced in a Machine Learning algorithm. We assess the performance of our pipeline, and in parallel, we compare the different representation methods on popular benchmark datasets. This work is a first step toward both an easy and ready-to-use pipeline for data classification using persistent homology and Machine Learning, ...

Background: Complex diseases may have multiple pathways leading to disease. E.g. coronary artery disease evolves from arterial damage to their epithelial layers, but has multiple causal pathways. More challenging, those pathways are highly correlated within metabolic syndrome. The challenge is to identify specific clusters of phenotype characteristics (composite phenotypes) that may reflect these different etiologies. Further, GWAS seeking to identify SNPs satisfying multiple composite phenotype descriptions allows for lower false positive rates at lower α thresholds, allowing for the possibility of reducing false negatives. This may provide a window into the missing heritability problem. Methods: We identify significant phenotype patterns, and identify fuzzy redescriptions among those patterns using Jaccard distances. Further, we construct Vietoris-Rips complexes from the Jaccard distances and compute the persistent homology associated with those. The patterns comprising these topological features are identified as composite phenotpyes, whose genetic associations are explored with logistic regression applied to pathways and to GWAS. We identified several phenotypes that tended to be dominated by metabolic syndrome descriptions, and which were distinct among the combinations of metabolic syndrome conditions. Among SNPs marking the RAAS complex, various SNPs associated specifically with different groups of composite phenotypes, as well as distinguishing between the composite phenotypes and simple phenotypes. Each of these showed different genetic associations, namely rs6693954, rs762551, rs1378942, and rs1133323. GWAS identified SNPs that associated with composite phenotypes included rs12365545, rs6847235, and rs701319. Eighteen GWAS identified SNPs appeared in combinations supported in composite combinations with greater power than for any individual phenotype. Conclusions: We do find systematic associations among metabolic syndrome variates that show distinctive genetic association profiles. Further, the systematic characterization involves composite phenotype descriptions that allow for combined power of individual phenotype GWAS tests, yielding more significance for lower individual thresholds, permitting the exploration of SNPs that would otherwise show as false negatives.

Polydopamine has the capacity to adhere to a large variety of materials and this property offers the possibility to bind nanoparticles to solid surfaces. In this work, magnetite nanoparticles were deposited on gold substrates coated with polydopamine films. The aim of this work was to investigate the effects of the composition and morphology of the PDA layers on the sticking of magnetite nanoparticles. The polydopamine coating of gold surfaces was achieved by the oxidation of alkaline solutions of dopamine with various reaction times. The length of the reaction time to form PDA was expected to influence the composition and surface roughness of the PDA films. Magnetite nanoparticles were deposited on these polydopamine films by immersing the samples in aqueous dispersions of nanoparticles. The morphology at the nanometric scale and the composition of the surfaces before and after the deposition of magnetite nanoparticles were investigated by means of AFM and XPS. We found that the amount of magnetite nanoparticles on the surface did not vary monotonically with the reaction time of PDA formation, but it was at the minimum after 20 min of reaction. This behavior may be attributed to changes in the chemical composition of the coating layer with reaction time.

A basic task in signal analysis is to characterize data in a meaningful way for analysis and classification purposes. Time-Frequency transforms are powerful strategies for signal decomposition, and important recent generalizations have been achieved in the setting of frame theory. In parallel recent developments, tools from algebraic topology, traditionally developed in purely abstract settings, have provided new insights in applications to data analysis. In this report, we investigate some interactions of these tools, both theoretically and with numerical experiments in order to characterize signals and their corresponding adaptive frames. We explain basic concepts in persistent homology as an important new subfield of computational topology, as well as formulations of time-frequency analysis in frame theory. Our objective is to use persistent homology for constructing topological signatures of signals in the context of frame theory for classification and analysis purposes. The mai...

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications.

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications.

Persistent homology is a new tool from algebraic topology, showing until nowadays a lot of success when it comes to application in biology since this latest use metrics only for measuring similarities, Embedding the geometric details and focusing on the global shape is the key point making the success of persistent homology as an efficient topological data analysis tool. In this work we will be confirming the latest assumption (topology embeds geometry) by analyzing the structure of COILED SERINE which is a protein estimated to constitute 3-5 percent of the encoded residues in most genomes, and giving a substitute of the optimal characteristic distance that can be used in the flexibility-rigidity index, a classic method used to simulate molecule movements and flexible behavior, when it comes to atomic rigidity functions. We will also analyze interesting patterns in the binding site of the beta sheet generated from the pdb file 2JOX. We will be detecting and giving a simple descripti...

Motivated by the recent surge of criminal activities with cross-cryptocurrency trades, we introduce a new topological perspective to structural anomaly detection in dynamic multilayer networks. We postulate that anomalies in the underlying blockchain transaction graph that are composed of multiple layers are likely to also be manifested in anomalous patterns of the network shape properties. As such, we invoke the machinery of clique persistent homology on graphs to systematically and efficiently track evolution of the network shape and, as a result, to detect changes in the underlying network topology and geometry. We develop a new persistence summary for multilayer networks, called stacked persistence diagram, and prove its stability under input data perturbations. We validate our new topological anomaly detection framework in application to dynamic multilayer networks from the Ethereum Blockchain and the Ripple Credit Network, and demonstrate that our stacked PD approach substantially outperforms state-of-art techniques.

Motivated by the recent surge of criminal activities with cross-cryptocurrency trades, we introduce a new topological perspective to structural anomaly detection in dynamic multilayer networks. We postulate that anomalies in the underlying blockchain transaction graph that are composed of multiple layers are likely to also be manifested in anomalous patterns of the network shape properties. As such, we invoke the machinery of clique persistent homology on graphs to systematically and efficiently track evolution of the network shape and, as a result, to detect changes in the underlying network topology and geometry. We develop a new persistence summary for multilayer networks, called stacked persistence diagram, and prove its stability under input data perturbations. We validate our new topological anomaly detection framework in application to dynamic multilayer networks from the Ethereum Blockchain and the Ripple Credit Network, and demonstrate that our stacked PD approach substantially outperforms state-of-art techniques.

We propose a deterministic initialization of the Echo State Network reservoirs to ensure that the activation of its internal echo state representations reflects similar topological qualities of the input signal which should lead to a self-organizing reservoir. Human actions encoded as a multivariate time series signal are clustered before using the clustered nodes and interconnectivity matrices for initializing the S-ConvESN reservoirs. The capability of S-ConvESN is evaluated using several 3Dskeleton-based action recognition datasets.

Let M be a finitely generated bigraded module over the standard bigraded polynomial ring S = K[x 1 ,. .. , x m , y 1 ,. .. , y n ], and let Q = (y 1 ,. .. , y n). The local cohomology modules H k Q (M) are naturally bigraded, and the components H k Q (M) j = i H k Q (M) (i,j) are finitely generated graded K[x 1 ,. .. , x m ]modules. In this paper we study the regularity of H k Q (M) j , and show in several cases that reg H k Q (M) j is linearly bounded as a function of j.

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • We propose an automatic segmentation algorithm for histological images. • The algorithm uses persistent homology in a computational topology framework. Robust detection and segmentation of cell nuclei in biomedical images based on a computational topology framework

Persistent homology is a tool from a set of methods called Topological data analysis, showing until nowadays a lot of success when it comes to application in biology since this latest uses metrics only for measuring similarities, Embedding the geometric details and focusing on the global shape is the key point making the success of persistent homology, this will be investigated in the paper since enormous work already done in the field and results seems to be endless, as an efficient topological data analysis tool. In this work we will be confirming the latest assumption (topology embeds geometry) by displaying the structure of COILED SERINE which is a protein estimated to constitute 3-5 percent of the encoded residues in most genomes, and giving a substitute of the optimal characteristic distance that can be used in the flexibility-rigidity index, a classic method used to simulate molecule movements and flexible behavior, when it comes to atomic rigidity functions. We will also analyze interesting patterns in the binding site of the beta sheet generated from the pdb file 2JOX. We will be detecting and giving a simple description of different patterns generated by using javaplex generating barcodes and linear statistical results as a summary statistics.

The rapid evolution of image processing equipment and techniques ensures the development of novel picture analysis methodologies. One of the most powerful yet computationally possible algebraic techniques for measuring the topological characteristics of functions is persistent homology. It's an algebraic invariant that can capture topological details at different spatial resolutions. Persistent homology investigates the topological features of a space using a set of sampled points, such as pixels. It can track the appearance and disappearance of topological features caused by changes in the nested space created by an operation known as filtration, in which a parameter scale, in our case the intensity of pixels, is increased to detect changes in the studied space over a range of varying scales. In addition, at the level of machine learning there were many studies and articles witnessing recently the combination between homological persistence and machine learning algorithms. On a...

In this paper, we demonstrate that algebraic topology can be used to perform 2D+t object detection. After the construction of a topological complex for a 2D+t image sequence, we build a nested sequence of cell complexes on which relative persistent homology is computed. The relative homology adds to “absolute” homology the computation of classes related to the first and last frames of the sequence. By identifying 2D chains with large life spans, the most persistent classes are extracted. This allows for the identification of the interesting parts in a sequence and for the detection of the movement of objects despite continuous deformations in the image domain. The results obtained on a synthetic image and on two real biomedical images with moving vesicles recorded by a quantitative phase time‐lapse technique show the potential of this method. Comparing the method with a newly developed tracking tool proves that the strength of this method is its independence from prior parameters.

Quantification and classification of protein structures, such as knotted proteins, often requires noise-free and complete data. Here, we develop a mathematical pipeline that systematically analyses protein structures. We showcase this geometric framework on proteins forming open-ended trefoil knots, and we demonstrate that the mathematical tool, persistent homology, faithfully represents their structural homology. This topological pipeline identifies important geometric features of protein entanglement and clusters the space of trefoil proteins according to their depth. Persistence landscapes quantify the topological difference between a family of knotted and unknotted proteins in the same structural homology class. This difference is localized and interpreted geometrically with recent advancements in systematic computation of homology generators. The topological and geometric quantification we find is robust to noisy input data, which demonstrates the potential of this approach in ...

In Mathematical Morphology (MM), connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) that represents the stability of the extrema of a Morse function. Since these two concepts seem to be closely related, in this paper we examine their relationship, and we prove that they are equal on n-D Morse functions, n ≥ 1. More exactly, pairing a minimum with a 1-saddle by dynamics or pairing the same 1-saddle with a minimum by persistence leads exactly to the same pairing, assuming that the critical values of the studied Morse function are unique. This result is a step further to show how much topological data analysis and mathematical morphology are related, paving the way for a more in-depth study of the relations between these two research fields.

In Mathematical Morphology (MM), dynamics are used to compute markers to proceed for example to watershed-based image decomposition. At the same time, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) and represents the stability of the extrema of a Morse function. Since these concepts are similar on Morse functions, we studied their relationship and we found, and proved, that they are equal on 1D Morse functions. Here, we propose to extend this proof to n-D, n ≥ 2, showing that this equality can be applied to n-D images and not only to 1D functions. This is a step further to show how much MM and MT are related.

We state in this paper a strong relation existing between Mathematical Morphology and Discrete Morse Theory when we work with 1D Morse functions. Specifically, in Mathematical Morphology, a classic way to extract robust markers for segmentation purposes, is to use the dynamics. On the other hand, in Discrete Morse Theory, a wellknown tool to simplify the Morse-Smale complexes representing the topological information of a Morse function is the persistence. We show that pairing by persistence is equivalent to pairing by dynamics. Furthermore, self-duality and injectivity of these pairings are proved.

In Mathematical Morphology (MM), connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) that represents the stability of the extrema of a Morse function. Since these two concepts seem to be closely related, in this paper we examine their relationship, and we prove that they are equal on n-D Morse functions, n ≥ 1. More exactly, pairing a minimum with a 1-saddle by dynamics or pairing the same 1-saddle with a minimum by persistence leads exactly to the same pairing, assuming that the critical values of the studied Morse function are unique. This result is a step further to show how much topological data analysis and mathematical morphology are related, paving the way for a more in-depth study of the relations between these two research fields.

• Statistical inference of persistent homology over 3D rock images predicts constitutive behaviors. • Principal component analysis and a penalized regression model computes structural characteristics of sample volumes. • Output is consistent with established sREV sizes, and fluid flow and transport properties, but requires low computational cost.

Current morphometric methods that comprehensively measure shape cannot compare the disparate leaf shapes found in seed plants and are sensitive to processing artifacts. We explore the use of persistent homology, a topological method applied as a filtration across simplicial complexes (or more simply, a method to measure topological features of spaces across different spatial resolutions), to overcome these limitations. The described method isolates subsets of shape features and measures the spatial relationship of neighboring pixel densities in a shape. We apply the method to the analysis of 182,707 leaves, both published and unpublished, representing 141 plant families collected from 75 sites throughout the world. By measuring leaves from throughout the seed plants using persistent homology, a defined morphospace comparing all leaves is demarcated. Clear differences in shape between major phylogenetic groups are detected and estimates of leaf shape diversity within plant families a...

Current morphometric methods that comprehensively measure shape cannot compare the disparate leaf shapes found in seed plants and are sensitive to processing artifacts. We explore the use of persistent homology, a topological method applied across the scales of a function, to overcome these limitations. The described method isolates subsets of shape features and measures the spatial relationship of neighboring pixel densities in a shape. We apply the method to the analysis of 182,707 leaves, both published and unpublished, representing 141 plant families collected from 75 sites throughout the world. By measuring leaves from throughout the seed plants using persistent homology, a defined morphospace comparing all leaves is demarcated. Clear differences in shape between major phylogenetic groups are detected and estimates of leaf shape diversity within plant families are made. This approach does not only predict plant family, but also the collection site, confirming phylogenetically i...

Scientific data has been growing in both size and complexity across the modern physical, engineering, life and social sciences. Spatial structure, for example, is a hallmark of many of the most important real-world complex systems, but its analysis is fraught with statistical challenges. Topological data analysis can provide a powerful computational window on complex systems. Here we present a framework to extend and interpret persistent homology summaries to analyse spatial data across multiple scales. We introduce hyperTDA, a topological pipeline that unifies local (e.g. geodesic) and global (e.g. Euclidean) metrics without losing spatial information, even in the presence of noise. Homology generators offer an elegant and flexible description of spatial structures and can capture the information computed by persistent homology in an interpretable way. Here the information computed by persistent homology is transformed into a weighted hypergraph, where hyperedges correspond to homology generators. We consider different choices of generators (e.g. matroid or minimal) and find that centrality and community detection are robust to either choice. We compare hyperTDA to existing geometric measures and validate its robustness to noise. We demonstrate the power of computing higher-order topological structures on spatial curves arising frequently in ecology, biophysics, and biology, but also in high-dimensional financial datasets. We find that hyperTDA can select between synthetic trajectories from the landmark 2020 AnDi challenge and quantifies movements of different animal species, even when data is limited.

We study the relationship between a notion of conjunction among conditional events, introduced in recent papers, and the notion of Frank t-norm. By examining different cases, in the setting of coherence, we show each time that the conjunction coincides with a suitable Frank t-norm. In particular, the conjunction may coincide with the Product t-norm, the Minimum t-norm, and Lukasiewicz t-norm. We show by a counterexample, that the prevision assessments obtained by Lukasiewicz t-norm may be not coherent. Then, we give some conditions of coherence when using Lukasiewicz t-norm.

We study the relationship between a notion of conjunction among conditional events, introduced in recent papers, and the notion of Frank t-norm. By examining different cases, in the setting of coherence, we show each time that the conjunction coincides with a suitable Frank t-norm. In particular, the conjunction may coincide with the Product t-norm, the Minimum t-norm, and Lukasiewicz t-norm. We show by a counterexample, that the prevision assessments obtained by Lukasiewicz t-norm may be not coherent. Then, we give some conditions of coherence when using Lukasiewicz t-norm.

Sparse systems are usually parameterized by a tuning parameter that determines the sparsity of the system. How to choose the right tuning parameter is a fundamental and difficult problem in learning the sparse system. In this paper, by treating the the tuning parameter as an additional dimension, persistent homological structures over the parameter space is introduced and explored. The structures are then further exploited in speeding up the computation using the proposed soft-thresholding technique. The topological structures are further used as multivariate features in the tensor-based morphometry (TBM) in characterizing white matter alterations in children who have experienced severe early life stress and maltreatment. These analyses reveal that stress-exposed children exhibit more diffuse anatomical organization across the whole white matter region.