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The challenge of selecting the most relevant features of a given dataset arises ubiquitously in data analysis and dimensionality reduction. However, features found to be of high importance for the entire dataset may not be relevant to subsets of interest, and vice versa. Given a feature selector and a fixed decomposition of the data into subsets, we describe a method for identifying selected features which are compatible with the decomposition into subsets. We achieve this by re-framing the problem of finding compatible features to one of finding sections of a suitable quiver representation. In order to approximate such sections, we then introduce a Laplacian operator for quiver representations valued in Hilbert spaces. We provide explicit bounds on how the spectrum of a quiver Laplacian changes when the representation and the underlying quiver are modified in certain natural ways. Finally, we apply this machinery to the study of peak-calling algorithms which measure chromatin accessibility in single-cell data. We demonstrate that eigenvectors of the associated quiver Laplacian yield locally and globally compatible features., Comment: 40 pages, 7 figures

We introduce Hades, an unsupervised algorithm to detect singularities in data. This algorithm employs a kernel goodness-of-fit test, and as a consequence it is much faster and far more scaleable than the existing topology-based alternatives. Using tools from differential geometry and optimal transport theory, we prove that Hades correctly detects singularities with high probability when the data sample lives on a transverse intersection of equidimensional manifolds. In computational experiments, Hades recovers singularities in synthetically generated data, branching points in road network data, intersection rings in molecular conformation space, and anomalies in image data.

A key object of study in stochastic topology is a random simplicial complex. In this work we study a multi-parameter random simplicial complex model, where the probability of including a $k$-simplex, given the lower dimensional structure, is fixed. This leads to a conditionally independent probabilistic structure. This model includes the Erd\H{o}s-R\'enyi random graph, the random clique complex as well as the Linial-Meshulam complex as special cases. The model is studied from both probabilistic and statistical points of view. We prove multivariate central limit theorems with bounds and known limiting covariance structure for the subcomplex counts and the number of critical simplices under a lexicographical acyclic partial matching. We use the CLTs to develop a goodness-of-fit test for this random model and evaluate its empirical performance. In order for the test to be applicable in practice, we also prove that the MLE estimators are asymptotically unbiased, consistent, uncorrelated and normally distributed.

We describe an algorithm to compute Whitney stratifications of real algebraic varieties. The basic idea is to first stratify the complexified version of the given real variety using conormal techniques, and then to show that the resulting stratifications admit a description using only real polynomials. This method also extends to stratification problems involving certain basic semialgebraic sets as well as real algebraic maps. One of the map stratification algorithms described here yields a new method for solving the real root classification problem., Comment: 25 pages, 4 figures. Sec 2 is new and the introduction is expanded. Comments still welcome!

We count singular vector tuples of a system of tensors assigned to the edges of a directed hypergraph. To do so, we study the generalisation of quivers to directed hypergraphs. Assigning vector spaces to the nodes of a hypergraph and multilinear maps to its hyperedges gives a hyperquiver representation. Hyperquiver representations generalise quiver representations (where all hyperedges are edges) and tensors (where there is only one multilinear map). The singular vectors of a hyperquiver representation are a compatible assignment of vectors to the nodes. We compute the dimension and degree of the variety of singular vectors of a sufficiently generic hyperquiver representation. Our formula specialises to known results that count the singular vectors and eigenvectors of a generic tensor., Comment: May 12 2023: Removed redundant page auto-generated by arXiv

The Harder-Narasimhan type of a quiver representation is a discrete invariant parameterised by a real-valued function (called a central charge) defined on the vertices of the quiver. In this paper, we investigate the strength and limitations of Harder-Narasimhan types for several families of quiver representations which arise in the study of persistence modules. We introduce the skyscraper invariant, which amalgamates the HN types along central charges supported at single vertices, and generalise the rank invariant from multiparameter persistence modules to arbitrary quiver representations. Our four main results are as follows: (1) we show that the skyscraper invariant is strictly finer than the rank invariant in full generality, (2) we characterise the set of complete central charges for zigzag (and hence, ordinary) persistence modules, (3) we extend the preceding characterisation to rectangle-decomposable multiparameter persistence modules of arbitrary dimension; and finally, (4) we show that although no single central charge is complete for nestfree ladder persistence modules, a finite set of central charges is complete., Comment: 28 pages, comments welcome!

Attempts to incorporate topological information in supervised learning tasks have resulted in the creation of several techniques for vectorizing persistent homology barcodes. In this paper, we study thirteen such methods. Besides describing an organizational framework for these methods, we comprehensively benchmark them against three well-known classification tasks. Surprisingly, we discover that the best-performing method is a simple vectorization, which consists only of a few elementary summary statistics. Finally, we provide a convenient web application which has been designed to facilitate exploration and experimentation with various vectorization methods.

We introduce a sheaf-theoretic stability condition for finite acyclic quivers. Our main result establishes that for representations of affine type $\widetilde{\mathbb{A}}$ quivers, there is a precise relationship between the associated Harder-Narasimhan filtration and the barcode of the periodic zigzag persistence module obtained by unwinding the underlying quiver., Comment: 27 pages, 1 figure, comments welcome!

The Conley index of an isolated invariant set is a fundamental object in the study of dynamical systems. Here we consider smooth functions on closed submanifolds of Euclidean space and describe a framework for inferring the Conley index of any compact, connected isolated critical set of such a function with high confidence from a sufficiently large finite point sample. The main construction of this paper is a specific index pair which is local to the critical set in question. We establish that these index pairs have positive reach and hence admit a sampling theory for robust homology inference. This allows us to estimate the Conley index, and as a direct consequence, we are also able to estimate the Morse index of any critical point of a Morse function using finitely many local evaluations., Comment: 33 pages, comments welcome

We construct an equivariant version of discrete Morse theory for simplicial complexes endowed with group actions. The key ingredient is a 2-categorical criterion for making acyclic partial matchings on the quotient space compatible with an overlaid complex of groups. We use the discrete flow category of any such compatible matching to build the corresponding Morse complex of groups. Our main result establishes that the development of the Morse complex of groups recovers the original simplicial complex up to equivariant homotopy equivalence., Comment: 31 pages, 6 figures

A common approach for describing classes of functions and probability measures on a topological space $\mathcal{X}$ is to construct a suitable map $\Phi$ from $\mathcal{X}$ into a vector space, where linear methods can be applied to address both problems. The case where $\mathcal{X}$ is a space of paths $[0,1] \to \mathbb{R}^n$ and $\Phi$ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized $\Phi$ for the case where $\mathcal{X}$ is a space of maps $[0,1]^d \to \mathbb{R}^n$ for any $d \in \mathbb{N}$, and show that the map $\Phi$ generalizes many of the desirable algebraic and analytic properties of the path signature to $d \ge 2$. The key ingredient to our approach is topological; in particular, our starting point is a generalisation of K-T Chen's path space cochain construction to the setting of cubical mapping spaces., Comment: 58 pages, comments welcome!

Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: (1) the vector of critical simplex counts attained by a lexicographical Morse matching, (2) the vector of simplex counts in the link of a fixed simplex, and (3) the vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein's method.

The barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special form. Here we describe a new algorithm for computing barcodes which also keeps track of, and outputs, such a change of basis. Our main result is an explicit characterisation of the group of transformations that sends one barcode basis to another. Armed with knowledge of the entire space of barcode bases, we are able to show that any map of persistence modules can be represented via a partial matching between bars provided that neither source nor target admits nested bars in its barcode. We also generalise the algorithm and results described above to work for zizag modules., Comment: 29 pages

Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher dimensional topological features of data, features to which graphs, encoding only pairwise relationships, remain oblivious. While attempts have been made to extend Graph Neural Networks (GNNs) to a simplicial complex setting, the methods do not inherently exploit, or reason about, the underlying topological structure of the network. We propose a graph convolutional model for learning functions parametrized by the $k$-homological features of simplicial complexes. By spectrally manipulating their combinatorial $k$-dimensional Hodge Laplacians, the proposed model enables learning topological features of the underlying simplicial complexes, specifically, the distance of each $k$-simplex from the nearest "optimal" $k$-th homology generator, effectively providing an alternative to homology localization., Comment: 9 pages, 5 figures

Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of that manifold with high confidence. The algorithm for this estimation is Local PCA, a local version of principal component analysis. Our results accommodate for noisy non-uniform data distribution with the noise that may vary across the manifold, and allow simultaneous estimation at multiple points. Crucially, all of the constants appearing in our bound are explicitly described. The proof uses a matrix concentration inequality to estimate covariance matrices and a Wasserstein distance bound for quantifying nonlinearity of the underlying manifold and non-uniformity of the probability measure., Comment: Main theorems rewritten. Introduction is written more compactly

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to L\^e and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via primary decomposition, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps., Comment: There is an error in the published version of the article (Found Comput Math, 2022) which has been fixed in this update. Section 3 is entirely new, but the downstream results Sections 4-6 remain largely the same. We have also updated the Runtimes and Complexity estimates in Section 7. The def. of the integral closure of an ideal has also been corrected

Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections, and are eigenvectors of an associated matrix pencil., Comment: 31 pages; v2: more examples, results added to section 5

We describe a framework for estimating Hilbert-Samuel multiplicities $e_XY$ for pairs of projective varieties $X \subset Y$ from finite point samples rather than defining equations. The first step involves proving that this multiplicity remains invariant under certain hyperplane sections which reduce $X$ to a point $p$ and $Y$ to a curve $C$. Next, we establish that $e_pC$ equals the Euler characteristic (and hence, the cardinality) of the complex link of $p$ in $C$. Finally, we provide explicit bounds on the number of uniform point samples needed (in an annular neighborhood of $p$ in $C$) to determine this Euler characteristic with high confidence., Comment: 16 pages, 7 figures. This is a major revision. There is a problem with the Lefschetz theorem from Sec 6 of v1: it applies not to the usual complex linking space but to a projective version thereof (which has a very different topology in general). We have removed the old Sec 6 and replaced it with a new theorem on inferring multiplicities from point samples with high confidence

This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can optimally stratify a given dataset into clusters, each of which is in turn well-approximable by a suitable submanifold of the ambient space. Since these approximating submanifolds might have different dimensions, we are able to detect non-manifold like singular regions in data even when none of the data points have been sampled from those singularities. We showcase this method by identifying the intersection of two surfaces in the 24-dimensional space of cyclo-octane conformations, and by locating all the self-intersections of a Henneberg minimal surface immersed in 3-dimensional space. Due to the local nature of the required topological computations, the algorithmic burden of performing such data stratification is readily distributable across several processors.

A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of the sheaf to the stalks of the cosheaf. In this note we describe how, given a bisheaf constructible (i.e., locally constant) with respect to a triangulation of its underlying space, one can explicitly determine the coarsest stratification of that space for which the bisheaf remains constructible., Comment: 10 pages; this is the Final Version which appeared in the Proceedings of the 2018 Abel Symposium on Topological Data Analysis

We introduce and analyze parallelizable algorithms to compress and accurately reconstruct finite simplicial complexes that have non-trivial automorphisms. The compressed data -- called a complex of groups -- amounts to a functor from (the poset of simplices in) the orbit space to the 2-category of groups, whose higher structure is prescribed by isomorphisms arising from conjugation. Using this functor, we show how to algorithmically recover the original complex up to equivariant simplicial isomorphism. Our algorithms are derived from generalizations (by Bridson-Haefliger, Carbone-Rips and Corson, among others) of the classical Bass-Serre theory for reconstructing group actions on trees., Comment: 20 pages, 9 figures; Ver 2 contains minor improvements and a new Corollary 4.7

We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations - barcode to path, path to tensor series - results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks., Comment: Additional experiment and further details. To appear in IEEE Transactions on Pattern Analysis and Machine Intelligence

We outline an algorithm to recover the canonical (or, coarsest) stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology of cellular neighborhoods across the underlying complex. The result is a nested sequence of categories, each containing all the cells as its set of objects, with the property that two cells are isomorphic in the last category if and only if they lie in the same canonical stratum. The entire process is amenable to efficient distributed computation., Comment: Final version, published in Foundations of Computational Mathematics

The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching $\mu$ on a finite regular CW complex $X$, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of $X$. Forman also proved the existence of a CW complex which is homotopy equivalent to $X$ and whose cells are in one-to-one correspondence with the critical cells of $\mu$, but the construction is ad hoc and does not have a combinatorial description. By relaxing the definition of Forman's gradient flows, we introduce the notion of flow paths, which contains enough information to reconstruct the homotopy type of $X$, while retaining a combinatorial description. The critical difference from Forman's gradient flows is the existence of a partial order on the set of flow paths, from which a $2$-category $C(\mu)$ is constructed. It is shown that the classifying space of $C(\mu)$ is homotopy equivalent to $X$ by using homotopy theory of $2$-categories. This result can be also regarded as a discrete analogue of the unpublished work of Cohen, Jones, and Segal on Morse theory in early 90's., Comment: 56 pages, 7 figures. v2: minor revision based on referee's comments

The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion which guarantees the extensibility of non-expansive maps into this space across embeddings of the domain to larger ambient metric spaces. Our coherence criterion is category-theoretic, allowing Kan extensions to provide the desired extensions. Our main construction gives an isometric embedding of a metric space into the metric space of persistence modules with values in the spacetime of this metric space. As a consequence of such "higher interpolation", it becomes possible to compare Vietoris-Rips and \v{C}ech complexes built within the space of persistence modules., Comment: 12 Pages, 2 Figures

Incidence relations among the cells of a regular CW complex produce a poset-enriched category of entrance paths whose classifying space is homotopy-equivalent to that complex. We show here that each acyclic partial matching (in the sense of discrete Morse theory) on the cells corresponds precisely to a homotopy-preserving localization of the associated entrance path category. Restricting attention further to the full localized subcategory spanned by critical cells, we obtain the discrete flow category whose classifying space is also shown to lie in the homotopy class of the original CW complex. This flow category forms a combinatorial and computable counterpart to the one described by Cohen, Jones and Segal in the context of smooth Morse theory., Comment: 30 pages, 6 figures. Final version, which appeared in the Journal of Pure and Applied Algebra

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., Comment: 24 pages, 14 figures

Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex., Comment: 19 pages, 1 Figure. Added Section 5.3

From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopy-equivalent to that manifold with high confidence. We describe a corresponding result for a Lipschitz-continuous function between two such manifolds. That is, we outline the construction of a simplicial map which recovers the induced maps on homotopy and homology groups with high confidence using only finite sampled data from the domain and range, as well as knowledge of the image of every point sampled from the domain. We provide explicit bounds on the size of the point samples required for such reconstruction in terms of intrinsic properties of the domain, the co-domain and the function. This reconstruction is robust to certain types of bounded sampling and evaluation noise., Comment: 15 pages, To Appear in the Journal of Computational Dynamics (2014)

We formally investigate some computational obstacles to tractability of computing the variety determined by K complex polynomials in N boolean variables. We show that using algebraic methods for solving combinatorial problems, the obstacles to tractability lies in the order of magnitude of asymmetries admitted by the given system of equations.