Search

The Euler characteristic transform (ECT) is a simple to define yet powerful representation of shape. The idea is to encode an embedded shape by tracking how the Euler characteristic, a simple integer-valued topological invariant, changes as the shape is built up in a particular direction. 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 key concepts, theoretical foundations, and available applications. [ABSTRACT FROM AUTHOR]

In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across a filtration. In this work, we extend this setting, and propose the use of bipath persistent homology, which can be used to study the persistence of topological features across a pair of filtrations connected at their ends, to compare the two filtrations. In this setting, interval-decomposability is guaranteed, and we provide an algorithm for computing persistence diagrams for bipath persistent homology and discuss the interpretation of bipath persistence diagrams. [ABSTRACT FROM AUTHOR]

Persistence modules and barcodes are used in symplectic topology to define various invariants of Hamiltonian diffeomorphisms, however numerical methods for computing these barcodes are not yet well developed. In this paper we define one such invariant called the generating function barcode of compactly supported Hamiltonian diffeomorphisms of R 2 n by applying Morse theory to generating functions quadratic at infinity associated to such Hamiltonian diffeomorphisms and provide an algorithm (i.e a finite sequence of explicit calculation steps) that approximates it. [ABSTRACT FROM AUTHOR]

Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the more general problem of computing the image under the homology functor of a continuous semi-algebraic map f : X → Y between closed and bounded semi-algebraic sets. For every fixed ℓ ≥ 0 we give an algorithm with singly exponential complexity that computes bases of the homology groups H i (X) , H i (Y) (with rational coefficients) and a matrix with respect to these bases of the induced linear maps H i (f) : H i (X) → H i (Y) , 0 ≤ i ≤ ℓ . We generalize this algorithm to more general (zigzag) diagrams of continuous semi-algebraic maps between closed and bounded semi-algebraic sets and give a singly exponential algorithm for computing the homology functors on such diagrams. This allows us to give an algorithm with singly exponential complexity for computing barcodes of semi-algebraic zigzag persistent homology in small dimensions. [ABSTRACT FROM AUTHOR]

We study the reach (in the sense of Federer) of the natural isometric embedding X ↪ W p (X) of X inside its p-Wasserstein space, where (X , dist) is a geodesic metric space. We prove that if a point x ∈ X can be joined to another point y ∈ X by two minimizing geodesics, then reach (x , X ⊂ W p (X)) = 0 . This includes the cases where X is a compact manifold or a non-simply connected one. On the other hand, we show that reach (X ⊂ W p (X)) = ∞ when X is a CAT(0) space. The infinite reach enables us to examine the regularity of the projection map. Furthermore, we replicate these findings by considering the isometric embedding X ↪ W ϑ (X) into an Orlicz–Wasserstein space, a generalization by Sturm of the classical Wasserstein space. Lastly, we establish the nullity of the reach for the isometric embedding of X ↪ Dgm ∞ , the space of persistence diagrams equipped with the bottleneck distance. [ABSTRACT FROM AUTHOR]

Topological data analysis (TDA) is an active field of mathematics for quantifying shape in complex data. Standard methods in TDA such as persistent homology (PH) are typically focused on the analysis of data consisting of a single entity (e.g., cells or molecular species). However, state-of-the-art data collection techniques now generate exquisitely detailed multispecies data, prompting a need for methods that can examine and quantify the relations among them. Such heterogeneous data types arise in many contexts, ranging from biomedical imaging, geospatial analysis, to species ecology. Here, we propose two methods for encoding spatial relations among different data types that are based on Dowker complexes and Witness complexes. We apply the methods to synthetic multispecies data of a tumor microenvironment and analyze topological features that capture relations between different cell types, e.g., blood vessels, macrophages, tumor cells, and necrotic cells. We demonstrate that relational topological features can extract biological insight, including the dominant immune cell phenotype (an important predictor of patient prognosis) and the parameter regimes of a data-generating model. The methods provide a quantitative perspective on the relational analysis of multispecies spatial data, overcome the limits of traditional PH, and are readily computable. [ABSTRACT FROM AUTHOR]

We develop sheaf-theoretic methods to deal with non-smooth objects in symplectic geometry. We show the completeness of a derived category of sheaves with respect to the interleaving distance and construct a sheaf quantization of a Hamiltonian homeomorphism. We also develop Lusternik–Schnirelmann theory in the microlocal theory of sheaves. With these new sheaf-theoretic methods, we prove an Arnold-type theorem for the image of a compact exact Lagrangian submanifold under a Hamiltonian homeomorphism. [ABSTRACT FROM AUTHOR]

This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of S 1 ) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space using persistent homology. Under fairly mild conditions, we show that such a loop either terminates a 1-dimensional homology class or gives rise to a 2-dimensional homology class in persistent homology. The main tool in this detection technique are selective Rips complexes, new custom made complexes that function as an appropriate combinatorial lens for persistent homology to detect the above mentioned loops. The main argument is based on a new concept of a local winding number, which turns out to be an invariant of certain homology classes. [ABSTRACT FROM AUTHOR]

Metrics of interest in topological data analysis (TDA) are often explicitly or implicitly in the form of an interleaving distance d I between poset maps (i.e. order-preserving maps), e.g. the Gromov-Hausdorff distance between metric spaces can be reformulated in this way. We propose a representation of a poset map F : P → Q as a join (i.e. supremum) ⋁ b ∈ B F b of simpler poset maps F b (for a join dense subset B ⊂ Q ) which in turn yields a decomposition of d I into a product metric. The decomposition of d I is simple, but its ramifications are manifold: (1) We can construct a geodesic path between any poset maps F and G with d I (F , G) < ∞ by assembling geodesics between all F b s and G b s via the join operation. This construction generalizes at least three constructions of geodesic paths that have appeared in the literature. (2) We can extend the Gromov-Hausdorff distance to a distance between simplicial filtrations over an arbitrary poset with a flow, preserving its universality and geodesicity. (3) We can clarify equivalence between several known metrics on multiparameter hierarchical clusterings. (4) We can illuminate the relationship between the erosion distance by Patel and the graded rank function by Betthauser, Bubenik, and Edwards, which in turn takes us to an interpretation on the representation ⋁ b F b as a generalization of persistence landscapes and graded rank functions. [ABSTRACT FROM AUTHOR]

Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. This relies on another contribution of this paper, which eliminates the need to examine a factorial number of permutations of simplices with the same value. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude. [ABSTRACT FROM AUTHOR]

This is the first of a series of papers that develop a systematic bridge between constructions in discrete mathematics and the corresponding continuous analogs. In this paper, we establish an equivalence between Forman's discrete Morse theory on a simplicial complex and the continuous Morse theory (in the sense of any known non-smooth Morse theory) on the associated order complex via the Lovász extension. Furthermore, we propose a new version of the Lusternik–Schnirelman category on abstract simplicial complexes to bridge the classical Lusternik–Schnirelman theorem and its discrete analog on finite complexes. More generally, we can suggest a discrete Morse theory on hypergraphs by employing piecewise-linear (PL) Morse theory and Lovász extension, hoping to provide new tools for exploring the structure of hypergraphs. [ABSTRACT FROM AUTHOR]

Identifying and representing clusters in time-varying network data is of particular importance when studying collective behaviors emerging in nature, in mobile device networks or in social networks. Based on combinatorial, categorical, and persistence theoretic viewpoints, we establish a stable functorial pipeline for the summarization of the evolution of clusters in a time-varying network. We first construct a complete summary of the evolution of clusters in a given time-varying network over a set of entities X of which takes the form of a formigram. This formigram can be understood as a certain Reeb graph R which is labeled by subsets of X. By applying Möbius inversion to the formigram in two different manners, we obtain two dual notions of diagram: the maximal group diagram and the persistence clustergram, both of which are in the form of an 'annotated' barcode. The maximal group diagram consists of time intervals annotated by their corresponding maximal groups — a notion due to Buchin et al., implying that we recognize the notion of maximal groups as a special instance of generalized persistence diagram by Patel. On the other hand, the persistence clustergram is mostly obtained by annotating the intervals in the zigzag barcode of the Reeb graph R with certain merging/disbanding events in the given time-varying network. We show that both diagrams are complete invariants of formigrams (or equivalently of trajectory grouping structure by Buchin et al.) and thus contain more information than the Reeb graph R . [ABSTRACT FROM AUTHOR]

We provide novel lower bounds on the Betti numbers of Vietoris–Rips complexes of hypercube graphs of all dimensions and at all scales. In more detail, let Q n be the vertex set of 2 n vertices in the n-dimensional hypercube graph, equipped with the shortest path metric. Let VR (Q n ; r) be its Vietoris–Rips complex at scale parameter r ≥ 0 , which has Q n as its vertex set, and all subsets of diameter at most r as its simplices. For integers r < r ′ the inclusion VR (Q n ; r) ↪ VR (Q n ; r ′) is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces VR (Q n ; r) . We provide lower bounds on the ranks of homology groups of VR (Q n ; r) . For example, using cross-polytopal generators, we prove that the rank of H 2 r - 1 (VR (Q n ; r)) is at least 2 n - (r + 1) n r + 1 . We also prove a version of homology propagation: if q ≥ 1 and if p is the smallest integer for which rank H q (VR (Q p ; r)) ≠ 0 , then rank H q (VR (Q n ; r)) ≥ ∑ i = p n 2 i - p i - 1 p - 1 · rank H q (VR (Q p ; r)) for all n ≥ p . When r ≤ 3 , this result and variants thereof provide tight lower bounds on the rank of H q (VR (Q n ; r)) for all n, and for each r ≥ 4 we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each r ≥ 2 , the homology groups of VR (Q n ; r) for n ≥ 2 r + 1 contain propagated homology not induced by the initial cross-polytopal generators. [ABSTRACT FROM AUTHOR]

In topological data analysis (TDA), persistence diagrams (PDs) have been a successful tool. To compare them, Wasserstein and bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for PDs as a generalization of the bottleneck distance. These metrics satisfy a stability result and can be used to bound Wasserstein metrics from above and from below. We provide algorithms to compute the newly introduced quantities and end with an discussion about experiments. [ABSTRACT FROM AUTHOR]

We introduce a general definition of hybrid transforms for constructible functions. These are integral transforms combining Lebesgue integration and Euler calculus. Lebesgue integration gives access to well-studied kernels and to regularity results, while Euler calculus conveys topological information and allows for compatibility with operations on constructible functions. We conduct a systematic study of such transforms and introduce two new ones: the Euler–Fourier and Euler–Laplace transforms. We show that the first has a left inverse and that the second provides a satisfactory generalization of Govc and Hepworth's persistent magnitude to constructible sheaves, in particular to multi-parameter persistent modules. Finally, we prove index-theoretic formulae expressing a wide class of hybrid transforms as generalized Euler integral transforms. This yields expectation formulae for transforms of constructible functions associated with (sub)level-sets persistence of random Gaussian filtrations. [ABSTRACT FROM AUTHOR]

The Čech and Rips constructions of persistent homology are stable with respect to perturbations of the input data. However, neither is robust to outliers, and both can be insensitive to topological structure of high-density regions of the data. A natural solution is to consider 2-parameter persistence. This paper studies the stability of 2-parameter persistent homology: we show that several related density-sensitive constructions of bifiltrations from data satisfy stability properties accommodating the addition and removal of outliers. Specifically, we consider the multicover bifiltration, Sheehy's subdivision bifiltrations, and the degree bifiltrations. For the multicover and subdivision bifiltrations, we get 1-Lipschitz stability results closely analogous to the standard stability results for 1-parameter persistent homology. Our results for the degree bifiltrations are weaker, but they are tight, in a sense. As an application of our theory, we prove a law of large numbers for subdivision bifiltrations of random data. [ABSTRACT FROM AUTHOR]

We introduce a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology. From a zigzag filtration of complexes (X i) , we introduce a zigzag Morse filtration whose complexes (A i) are Morse reductions of the original complexes (X i) , and we prove that they both have same persistent homology. This zigzag Morse filtration generalizes the filtered Morse complex of Mischaikow and Nanda Mischaikow and Nanda (Discrete Comput Geom 50(2):330–353, 2013), defined for standard persistence. The maps in the zigzag Morse filtration are forward and backward inclusions, as is standard in zigzag persistence, as well as a new type of map inducing non trivial changes in the boundary operator of the Morse complex. We study in details this last map, and design algorithms to compute the update both at the complex level and at the homology matrix level when computing zigzag persistence. The key point of our construction is that it does not require any knowledge of past and future maps of the input filtration. We deduce an algorithm to compute the zigzag persistence of a filtration that depends mostly on the number of critical cells of the complexes, and show experimentally that it performs better in practice. [ABSTRACT FROM AUTHOR]

We examine the homotopy types of Vietoris–Rips complexes on certain finite metric spaces at scale 2. We consider the collections of subsets of [ m ] = { 1 , 2 , ... , m } equipped with symmetric difference metric d, specifically, F n m , F n m ∪ F n + 1 m , F n m ∪ F n + 2 m , and F ⪯ A m . Here F n m is the collection of size n subsets of [m] and F ⪯ A m is the collection of subsets ⪯ A where ⪯ is a total order on the collections of subsets of [m] and A ⊆ [ m ] (see the definition of ⪯ in Sect. 1). We prove that the Vietoris–Rips complexes V R (F n m , 2) and V R (F n m ∪ F n + 1 m , 2) are either contractible or homotopy equivalent to a wedge sum of S 2 's; also, the complexes V R (F n m ∪ F n + 2 m , 2) and V R (F ⪯ A m , 2) are either contractible or homotopy equivalent to a wedge sum of S 3 's. We provide inductive formulae for these homotopy types extending the result of Barmak about the independence complexes of Kneser graphs KG 2 , k and the result of Adamaszek and Adams about Vietoris–Rips complexes of hypercube graphs with scale 2. [ABSTRACT FROM AUTHOR]

Common AI music composition algorithms train a machine by feeding a set of music pieces. This approach is a blackbox optimization, i.e. the underlying composition algorithm is, in general, unknown to users. In this paper, we present a method of machine composition that teaches a machine the compositional principles embedded in the music using the concept of overlap matrix. In (Tran Mai Lan, Changbom Park & Jae-Hun Jung (2023) Topological data analysis of Korean music in Jeongganbo: a cycle structure, Journal of Mathematics and Music, DOI: 10.1080/17459737.2022.2164626), a type of Korean music called Dodeuri music has been analysed using topological data analysis (TDA). To apply TDA, the music data is first reconstructed as a graph. Through TDA on the constructed graph, a unique set of cycles is found. The overlap matrix lets us visualize how those cycles are interconnected in music. We explain how we use the overlap matrix for machine composition. The overlap matrix is suitable for algorithmic composition and also provides seed music to train an artificial neural network. [ABSTRACT FROM AUTHOR]

We develop a novel method to represent a weighted directed graph as a finite metric space and then use persistent homology to extract useful features. We apply this method to weighted directed graphs obtained from pitch transitions information of a given musical fragment and use these techniques to the quantitative study of stylistic trends. As a first illustration, we analyse a selection of string quartets by Haydn, Mozart and Beethoven and discuss possible implications of our results in terms of different approaches by these composers to stylistic exploration and variety. We observe that Haydn is stylistically the most conservative, followed by Mozart, while Beethoven is the most innovative. Finally we also compare the variability of different genres, namely minuets, allegros, prestos, and adagios, by a given composer and conclude that the minuet is the most stable form of the string quartet movements. [ABSTRACT FROM AUTHOR]

In the present article we describe and discuss a framework for applying different topological data analysis (TDA) techniques to a music fragment given as a score in traditional Western notation. We first consider different sets of points in Euclidean spaces of different dimensions that correspond to musical events in the score, and obtain their persistent homology features. Then we introduce two families of simplicial complexes that can be associated with chord sequences, and leverage homology to compute their salient features. Finally, we show the results of applying the described methods to the analysis and stylistic comparison of fragments from three Brandenburg Concertos by J.S. Bach and two pieces from the Graffiti series by Mexican composer Armando Luna. [ABSTRACT FROM AUTHOR]

The notion of generalized rank in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. However, its efficient computation has not yet been studied in the literature. We show that the generalized rank over a finite interval I of a Z 2 -indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank of M over I by computing the barcode of the zigzag module obtained by restricting to that path. If M is the homology of a bifiltration F of t simplices (while accounting for multi-criticality) and I consists of t points, this computation takes O (t ω) time where ω ∈ [ 2 , 2.373) is the exponent of matrix multiplication. We apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M, determine whether M is interval decomposable and, if so, compute all intervals supporting its indecomposable summands. [ABSTRACT FROM AUTHOR]

Motivated by topological data analysis, we study in this article certain notions of "tameness" for modules over posets. In particular, we show that after adding infinitary points the so called finitely determined modules become finitely presented. [ABSTRACT FROM AUTHOR]

We consider persistent homology obtained by applying homology to the open Rips filtration of a compact metric space (X, d). We show that each decrease in zero-dimensional persistence and each increase in one-dimensional persistence is induced by local minima of the distance function d. When d attains local minimum at only finitely many pairs of points, we prove that each above mentioned change in persistence is induced by a specific critical edge in Rips complexes, which represents a local minimum of d. We use this fact to develop a theory (including interpretation) of critical edges of persistence. The obtained results include upper bounds for the rank of one-dimensional persistence and a corresponding reconstruction result. Of potential computational interest is a simple geometric criterion recognizing local minima of d that induce a change in persistence. We conclude with a proof that each locally isolated minimum of d can be detected through persistent homology with selective Rips complexes. The results of this paper offer the first interpretation of critical scales of persistent homology (obtained via Rips complexes) for general compact metric spaces. [ABSTRACT FROM AUTHOR]

We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter persistence in topological data analysis. Our indecomposables of interest belong to the so-called interval modules, which by definition are indicator representations of intervals in the poset. While the whole class of interval modules does not admit such a local characterization, we show that the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest subclass to do so. [ABSTRACT FROM AUTHOR]

We introduce a novel gradient descent algorithm refining the well-known Gradient Sampling algorithm on the class of stratifiably smooth objective functions, which are defined as locally Lipschitz functions that are smooth on some regular pieces—called the strata—of the ambient Euclidean space. On this class of functions, our algorithm achieves a sub-linear convergence rate. We then apply our method to objective functions based on the (extended) persistent homology map computed over lower-star filters, which is a central tool of Topological Data Analysis. For this, we propose an efficient exploration of the corresponding stratification by using the Cayley graph of the permutation group. Finally, we provide benchmarks and novel topological optimization problems that demonstrate the utility and applicability of our framework. [ABSTRACT FROM AUTHOR]

Jeongganbo is a unique music representation invented by Sejong the Great. Contrary to the Western music notation, the pitch of each note is encrypted and the length is visualized directly in a matrix form. We use topological data analysis (TDA) to analyze the Korean music written in Jeongganbo for Suyeonjang, Songuyeo, and Taryong, those well-known pieces played among noble community. We define the nodes of each music with pitch and length and transform the music into a graph with the distance between the nodes defined as their adjacent occurrence rate. The graph homology is investigated by TDA. We identify cycles of each music and show how those cycles are interconnected. We found that the cycles of Suyeonjang and Songuyeo, categorized as a special type of cyclic music, frequently overlap each other in the music, while those of Taryong, which does not belong to the same class, appear only individually. [ABSTRACT FROM AUTHOR]