On the geometry of combinatorial manifolds

Penna, Michael A.

doi:10.2140/pjm.1978.77.499

Cited by 2 publications

(2 citation statements)

References 3 publications

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“…We found this definition in [20]. Note that some definitions of combinatorial manifolds require that the link of each vertex be simplicially isomorphic to S d , which is a stronger condition.…”

Section: A Particularly Illustrative (Though Not Exhaustive) Collecti...mentioning

confidence: 99%

Extreme Nonnegative Quadratics over Stanley Reisner Varieties

Shu

2021

Preprint

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We consider the convex geometry of the cone of nonnegative quadratics over Stanley-Reisner varieties. Stanley-Reisner varieties (which are unions of coordinate planes) are amongst the simplest real projective varieties, so this is potentially a starting point that can generalize to more complicated real projective varieties. This subject has some suprising connections to algebraic topology and category theory, which we exploit heavily in our work.These questions are also valuable in applied math, because they directly translate to questions about positive semidefinite (PSD) matrices. In particular, this relates to a long line of work concerning the extent to which it is possible to approximately check that a matrix is PSD by checking that some principle submatrices are PSD, or to check if a partial matrix can be approximately completed to full PSD matrix.We systematize both these practical and theoretical questions using a framework based on algebraic topology, category theory, and convex geometry. As applications of this framework we are able to classify the extreme nonnegative quadratics over many Stanley-Reisner varieties. We plan to follow these structural results with a paper that is more focused on quantitative questions about PSD matrix completion, which have applications in sparse semidefinite programming.

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Section: A Particularly Illustrative (Though Not Exhaustive) Collecti...mentioning

confidence: 99%

Extreme Nonnegative Quadratics over Stanley Reisner Varieties

Shu

2021

Preprint

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“…In theory, we assume the data given is a compact topological space X with an R d -valued function, f : X → R d , often denoted (X, f ). In practice, we assume the data we work with is a multivariate PL mapping f defined over a simplicial mesh; more restrictively (for easier exposition of our algorithms and proofs), we consider a generic, PL mapping f from a combinatorial manifold [20] to R d .…”

Section: Topological Notionsmentioning

confidence: 99%

Convergence between Categorical Representations of Reeb Space and Mapper

Munch,

Wang

2015

Preprint

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The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it compresses the components of the level sets of a multivariate mapping and obtains a summary representation of their relationships. A related construction called mapper, and a special case of the mapper construction called the Joint Contour Net have been shown to be effective in visual analytics. Mapper and JCN are intuitively regarded as discrete approximations of the Reeb space, however without formal proofs or approximation guarantees. An open question has been proposed by Dey et al. as to whether the mapper construction converges to the Reeb space in the limit.In this paper, we are interested in developing the theoretical understanding of the relationship between the Reeb space and its discrete approximations to support its use in practical data analysis. Using tools from category theory, we formally prove the convergence between the Reeb space and mapper in terms of an interleaving distance between their categorical representations. Given a sequence of refined discretizations, we prove that these approximations converge to the Reeb space in the interleaving distance; this also helps to quantify the approximation quality of the discretization at a fixed resolution.

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On the geometry of combinatorial manifolds

Cited by 2 publications

References 3 publications

Extreme Nonnegative Quadratics over Stanley Reisner Varieties

Extreme Nonnegative Quadratics over Stanley Reisner Varieties

Convergence between Categorical Representations of Reeb Space and Mapper

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