Local cohomology and stratification

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 Mathematic

Canonical stratifications along bisheaves

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 Analysi

Discrete Morse theory for computing cellular sheaf cohomology

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.

Conormal Spaces and Whitney Stratifications

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 correcte

Effective Whitney Stratification of Real Algebraic Varieties

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 certain algebraic maps. One of the map stratification algorithms described here yields a new method for solving the real root classification problem.Comment: 11 pages, 2 figures. Preliminary version --- we intend to add at least two more Sections. Comments welcome

Goodness-of-fit via Count Statistics in Dense Random Simplicial Complexes

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

Tangent space and dimension estimation with the Wasserstein distance

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

The Space of Barcode Bases for Persistence Modules

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 page