The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze data sets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. With a view toward establishing a general framework for the theory of GW-like distances, this paper considers a vast generalization of the notion of a metric measure space: for an arbitrary metric space $Z$, we define a $Z$-network to be a measure space endowed with a kernel valued in $Z$. We introduce a method for comparing $Z$-networks by defining a generalization of GW distance, which we refer to as $Z$-Gromov-Wasserstein ($Z$-GW) distance. This construction subsumes many previously known metrics and offers a unified approach to understanding their shared properties. This paper demonstrates that the $Z$-GW distance defines a metric on the space of $Z$-networks which retains desirable properties of $Z$, such as separability, completeness, and geodesicity. Many of these properties were unknown for existing variants of GW distance that fall under our framework. Our focus is on foundational theory, but our results also include computable lower bounds and approximations of the distance which will be useful for practical applications.

We introduce a notion of distance between supervised learning problems, which we call the Risk distance. This distance, inspired by optimal transport, facilitates stability results; one can quantify how seriously issues like sampling bias, noise, limited data, and approximations might change a given problem by bounding how much these modifications can move the problem under the Risk distance. With the distance established, we explore the geometry of the resulting space of supervised learning problems, providing explicit geodesics and proving that the set of classification problems is dense in a larger class of problems. We also provide two variants of the Risk distance: one that incorporates specified weights on a problem's predictors, and one that is more sensitive to the contours of a problem's risk landscape.

Abstract 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 $$\textbf{Z}^2$$ 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 $$t$$ t simplices (while accounting for multi-criticality) and I consists of $$t$$ t points, this computation takes $$O(t^\omega )$$ O ( t ω ) time where $$\omega \in [2,2.373)$$ 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 We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers $$k\ge 0$$ k ≥ 0 and $$n\ge 1$$ n ≥ 1 we consider the dimension k Vietoris–Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and denote them as $${\textbf{D}}_{n,k}^{\textrm{VR}}$$ D n , k VR . We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parameters n and k, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which $${\textbf{D}}_{4,1}^{\textrm{VR}}$$ D 4 , 1 VR fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a space X with cardinality $$2k+2$$ 2 k + 2 with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction.

We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the Möbius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing the meta-rank and meta-diagram of a 2-parameter module $M$ indexed by a bifiltration of $n$ simplices in $O(n^3)$ time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has $O(n^4)$ runtime. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of $M$ from $O(n^4)$ to $O(n^3)$. In addition, we define notions of erosion distance between meta-ranks and between meta-diagrams, and show that under these distances, meta-ranks and meta-diagrams are stable with respect to the interleaving distance. Lastly, the meta-diagram can be visualized in an intuitive fashion as a persistence diagram of diagrams, which generalizes the well-understood persistence diagram in the 1-parameter setting.

The (combinatorial) graph Laplacian is a fundamental object in the analysis of, and optimization on, graphs. Via a topological view, this operator can be extended to a simplicial complex K and therefore offers a way to perform "signal processing" on p-(co)chains of K. Recently, the concept of persistent Laplacian was proposed and studied for a pair of simplicial complexes K ↪ L connected by an inclusion relation, further broadening the use of Laplace-based operators. In this paper, we significantly expand the scope of the persistent Laplacian by generalizing it to a pair of weighted simplicial complexes connected by a weight preserving simplicial map f: K → L. Such a simplicial map setting arises frequently, e.g., when relating a coarsened simplicial representation with an original representation, or the case when the two simplicial complexes are spanned by different point sets, i.e. cases in which it does not hold that K ⊂ L. However, the simplicial map setting is much more challenging than the inclusion setting since the underlying algebraic structure is much more complicated. We present a natural generalization of the persistent Laplacian to the simplicial setting. To shed insight on the structure behind it, as well as to develop an algorithm to compute it, we exploit the relationship between the persistent Laplacian and the Schur complement of a matrix. A critical step is to view the Schur complement as a functorial way of restricting a self-adjoint positive semi-definite operator to a given subspace. As a consequence of this relation, we prove that the qth persistent Betti number of the simplicial map f: K → L equals the nullity of the qth persistent Laplacian Δ_q^{K,L}. We then propose an algorithm for finding the matrix representation of Δ_q^{K,L} which in turn yields a fundamentally different algorithm for computing the qth persistent Betti number of a simplicial map. Finally, we study the persistent Laplacian on simplicial towers under weight-preserving simplicial maps and establish monotonicity results for their eigenvalues.

We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the level of filtered chain complexes includes points with zero persistence which provide additional information to that present at homology level. The resulting invariant, called verbose barcode, which has a stronger discriminating power than the usual barcode, is proved to be stable under certain metrics that are sensitive to these ephemeral points. In some situations, we provide ways to compute such metrics between verbose barcodes. We also exhibit several examples of finite metric spaces with identical (standard) VR barcodes yet with different verbose VR barcodes thus confirming that these ephemeral points strengthen the standard VR barcode.

A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph into a large network. We propose two complementary MCMC algorithms for sampling random graph homomorphisms and establish bounds on their mixing times and the concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neighborhood sampling. Various time averages of the MCMC trajectory give us various computable observables, including well-known ones such as homomorphism density and average clustering coefficient and their generalizations. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut distance between networks. We provide various examples and simulations demonstrating our framework through synthetic networks. We also \commHL{demonstrate the performance of} our framework on the tasks of network clustering and subgraph classification on the Facebook100 dataset and on Word Adjacency Networks of a set of classic novels.

The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these rank invariants efficiently is a prelude to computing any of these derived structures efficiently. We show that the generalized rank over a finite interval $I$ of a $\mathbb{Z}^2$-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 over $I$ by computing the barcode of the zigzag module obtained by restricting the bifiltration inducing $M$ to that path. If the bifiltration and $I$ have at most $t$ simplices and points respectively, this computation takes $O(t^ω)$ time where $ω\in[2,2.373)$ is the exponent of matrix multiplication. Among others, 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 summands.

The Weisfeiler-Lehman (WL) test is a classical procedure for graph isomorphism testing. The WL test has also been widely used both for designing graph kernels and for analyzing graph neural networks. In this paper, we propose the Weisfeiler-Lehman (WL) distance, a notion of distance between labeled measure Markov chains (LMMCs), of which labeled graphs are special cases. The WL distance is polynomial time computable and is also compatible with the WL test in the sense that the former is positive if and only if the WL test can distinguish the two involved graphs. The WL distance captures and compares subtle structures of the underlying LMMCs and, as a consequence of this, it is more discriminating than the distance between graphs used for defining the state-of-the-art Wasserstein Weisfeiler-Lehman graph kernel. Inspired by the structure of the WL distance we identify a neural network architecture on LMMCs which turns out to be universal w.r.t. continuous functions defined on the space of all LMMCs (which includes all graphs) endowed with the WL distance. Finally, the WL distance turns out to be stable w.r.t. a natural variant of the Gromov-Wasserstein (GW) distance for comparing metric Markov chains that we identify. Hence, the WL distance can also be construed as a polynomial time lower bound for the GW distance which is in general NP-hard to compute.

We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.

An augmented metric space (X, d_X, f_X) is a metric space (X, d_X) equipped with a function f_X: X → ℝ. It arises commonly in practice, e.g, a point cloud X in ℝ^d where each point x∈ X has a density function value f_X(x) associated to it. Such an augmented metric space naturally gives rise to a 2-parameter filtration. However, the resulting 2-parameter persistence module could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of a 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space. Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane. We show that the fibered barcode, the fibered merge tree, and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X) can all be efficiently computed once the elder-rule-staircode is given. Furthermore, for certain special cases, this staircode corresponds exactly to the set of indecomposables of the zeroth homology of the 2-parameter filtration. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n²log n) time, which can be improved to O(n²α(n)) if X is from a fixed dimensional Euclidean space ℝ^d, where α(n) is the inverse Ackermann function.

In this paper, we consider the problem of projecting networks onto metric spaces. Networks are structures that encode relationships between pairs of elements or nodes. However, these relationships can be independent of each other, and need not be defined for every pair of nodes. This is in contrast to a metric space, which requires that a distance between every pair of elements in the space be defined. To understand how to project networks onto metric spaces, we take an axiomatic approach: we first state two axioms for projective maps from the set of all networks to the set of finite metric spaces, then show that only one projection satisfies these requirements. The developed technique is shown to be an effective method for finding approximate solutions to combinatorial optimization problems. Finally, we illustrate the use of metric trees for efficient search in projected networks.

We introduce the Wasserstein transform, a method for enhancing and denoising datasets defined on general metric spaces. The construction draws inspiration from Optimal Transportation ideas. We establish the stability of our method under data perturbation and, when the dataset is assumed to be Euclidean, we also exhibit a precise connection between the Wasserstein transform and the mean shift family of algorithms. We then use this connection to prove that mean shift also inherits stability under perturbations. We study the performance of the Wasserstein transform method on different datasets as a preprocessing step prior to clustering and classification tasks.

While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We extend a construction of homology of digraphs due to Grigoryan, Lin, Muranov and Yau to the persistent framework. The result, which we call persistent path homology or PPH, encodes a rich level of detail about the asymmetric structure of the input directed network. For example, we prove that PPH identifies a class of directed cyclic networks as directed analogues of the circle. In general, PPH produces signatures that differ from natural extensions of Rips or Čech persistence to the directed setting, but we prove that PPH agrees with Čech persistence on symmetric spaces. Additionally, we prove that PPH agrees with Čech persistence on directed networks satisfying a local condition that we call square-freeness. We prove stability of PPH by utilizing a separate theory of homotopy of digraphs that is compatible with path homology. Finally, we study computational aspects of PPH, and derive an algorithm showing that over field coefficients, computing PPH requires the same worst case running time as standard persistent homology.

Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called nerve of a cover of the domain. In this paper, we aim to analyze the topological information encoded in these structures in order to provide better understanding of these structures and facilitate their practical usage. More specifically, we show that the one-dimensional homology of the nerve complex N(U) of a path-connected cover U of a domain X cannot be richer than that of the domain X itself. Intuitively, this result means that no new H_1-homology class can be "created" under a natural map from X to the nerve complex N(U). Equipping X with a pseudometric d, we further refine this result and characterize the classes of H_1(X) that may survive in the nerve complex using the notion of size of the covering elements in U. These fundamental results about nerve complexes then lead to an analysis of the H_1-homology of Reeb spaces, mappers and multiscale mappers. The analysis of H_1-homology groups unfortunately does not extend to higher dimensions. Nevertheless, by using a map-induced metric, establishing a Gromov-Hausdorff convergence result between mappers and the domain, and interleaving relevant modules, we can still analyze the persistent homology groups of (multiscale) mappers to establish a connection to Reeb spaces.

Abstract Recent efforts in the area of joint object matching approach the problem by taking as input a set of pairwise maps, which are then jointly optimized across the whole collection so that certain accuracy and consistency criteria are satisfied. One natural requirement is cycle‐consistency—namely the fact that map composition should give the same result regardless of the path taken in the shape collection. In this paper, we introduce a novel approach to obtain consistent matches without requiring initial pairwise solutions to be given as input. We do so by optimizing a joint measure of metric distortion directly over the space of cycle‐consistent maps; in order to allow for partially similar and extra‐class shapes, we formulate the problem as a series of quadratic programs with sparsity‐inducing constraints, making our technique a natural candidate for analysing collections with a large presence of outliers. The particular form of the problem allows us to leverage results and tools from the field of evolutionary game theory. This enables a highly efficient optimization procedure which assures accurate and provably consistent solutions in a matter of minutes in collections with hundreds of shapes.

We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces. Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any n-point quasimetric space supported on a graph of treewidth t admits a random embedding into quasiultrametric spaces with distortion O(t*log^2(n)), where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain t*log^{O(1)}(n)-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on n-vertex graphs of treewidth t, with running time polynomial in both n and t. The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph G into graphs that exclude G as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting.

Networks which show the relationships within and between complex systems are key tools in a variety of current scientific areas. A central aim in network analysis is to find a suitable metric for network similarity and comparison. We propose a definition for the space of all networks, and show that our definition leads to a natural and meaningful notion of distance between networks. We discuss the computational complexity involved in computing our network distance, and develop lower bounds by using invariants of networks that are significantly simpler to compute. By constructing a wide range of explicit examples, we show that these lower bounds are effective in distinguishing between networks. We describe multiple invariants and prove that all of them are stable in a quantitative sense.

Estimating optimal phylogenetic trees or hierarchical clustering trees from metric data is an important problem in evolutionary biology and data analysis. Intuitively, the goodness-of-fit of a metric space to a tree depends on its inherent treeness, as well as other metric properties such as intrinsic dimension. Existing algorithms for embedding metric spaces into tree metrics provide distortion bounds depending on cardinality. Because cardinality is a simple property of any set, we argue that such bounds do not fully capture the rich structure endowed by the metric. We consider an embedding of a metric space into a tree proposed by Gromov. By proving a stability result, we obtain an improved additive distortion bound depending only on the hyperbolicity and doubling dimension of the metric. We observe that Gromov's method is dual to the well-known single linkage hierarchical clustering (SLHC) method. By means of this duality, we are able to transport our results to the setting of SLHC, where such additive distortion bounds were previously unknown.

Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. Mapper, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map defined on the data, and produces a summary of the data by using a cover of the codomain of the map. This cover, via a pullback operation to the domain, produces a simplicial complex connecting the data points. The resulting view of the data through a cover of the codomain offers flexibility in analyzing the data. However, it offers only a view at a fixed scale at which the cover is constructed. Inspired by the concept, we explore a notion of a tower of covers which induces a tower of simplicial complexes connected by simplicial maps, which we call multiscale mapper. We study the resulting structure, and design practical algorithms to compute its persistence diagrams efficiently. Specifically, when the domain is a simplicial complex and the map is a real-valued piecewise-linear function, the algorithm can compute the exact persistence diagram only from the 1-skeleton of the input complex. For general maps, we present a combinatorial version of the algorithm that acts only on vertex sets connected by the 1-skeleton graph, and this algorithm approximates the exact persistence diagram thanks to a stability result that we show to hold.

This paper introduces hierarchical quasi-clustering methods, a generalization of hierarchical clustering for asymmetric networks where the output structure preserves the asymmetry of the input data. We show that this output structure is equivalent to a finite quasi-ultrametric space and study admissibility with respect to two desirable properties. We prove that a modified version of single linkage is the only admissible quasi-clustering method. Moreover, we show stability of the proposed method and we establish invariance properties fulfilled by it. Algorithms are further developed and the value of quasi-clustering analysis is illustrated with a study of internal migration within United States.

We present an axiomatic construction of hierarchical clustering in asymmetric networks where the dissimilarity from node a to node b is not necessarily equal to the dissimilarity from node b to node a. The theory is built on the axioms of value and transformation which encode desirable properties common to any clustering method. Two hierarchical clustering methods that abide to these axioms are derived: reciprocal and nonreciprocal clustering. We further show that any clustering method that satisfies the axioms of value and transformation lies between reciprocal and nonreciprocal clustering in a well defined sense. We apply this theory to the formation of circles of trust in social networks.

An animal's ability to navigate through space rests on its ability to create a mental map of its environment. The hippocampus is the brain region centrally responsible for such maps, and it has been assumed to encode geometric information (distances, angles). Given, however, that hippocampal output consists of patterns of spiking across many neurons, and downstream regions must be able to translate those patterns into accurate information about an animal's spatial environment, we hypothesized that 1) the temporal pattern of neuronal firing, particularly co-firing, is key to decoding spatial information, and 2) since co-firing implies spatial overlap of place fields, a map encoded by co-firing will be based on connectivity and adjacency, i.e., it will be a topological map. Here we test this topological hypothesis with a simple model of hippocampal activity, varying three parameters (firing rate, place field size, and number of neurons) in computer simulations of rat trajectories in three topologically and geometrically distinct test environments. Using a computational algorithm based on recently developed tools from Persistent Homology theory in the field of algebraic topology, we find that the patterns of neuronal co-firing can, in fact, convey topological information about the environment in a biologically realistic length of time. Furthermore, our simulations reveal a "learning region" that highlights the interplay between the parameters in combining to produce hippocampal states that are more or less adept at map formation. For example, within the learning region a lower number of neurons firing can be compensated by adjustments in firing rate or place field size, but beyond a certain point map formation begins to fail. We propose that this learning region provides a coherent theoretical lens through which to view conditions that impair spatial learning by altering place cell firing rates or spatial specificity.

Abstract A common operation in many geometry processing algorithms consists of finding correspondences between pairs of shapes by finding structure‐preserving maps between them. A particularly useful case of such maps is isometries, which preserve geodesic distances between points on each shape. Although several algorithms have been proposed to find approximately isometric maps between a pair of shapes, the structure of the space of isometries is not well understood. In this paper, we show that under mild genericity conditions, a single correspondence can be used to recover an isometry defined on entire shapes, and thus the space of all isometries can be parameterized by one correspondence between a pair of points. Perhaps surprisingly, this result is general, and does not depend on the dimensionality or the genus, and is valid for compact manifolds in any dimension. Moreover, we show that both the initial correspondence and the isometry can be recovered efficiently in practice. This allows us to devise an algorithm to find intrinsic symmetries of shapes, match shapes undergoing isometric deformations, as well as match partial and incomplete models efficiently.

Abstract We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on the persistence diagrams of suitable filtrations built on top of these spaces. We prove the stability of our signatures under Gromov‐Hausdorff perturbations of the spaces. We also extend these results to metric spaces equipped with measures. Our signatures are well‐suited for the study of unstructured point cloud data, which we illustrate through an application in shape classification.