Related papers: Sheaf-Theoretic Stratification Learning from Geome…
Given a stratified variety X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of "parity sheaves", which are objects in the constructible derived category of sheaves with coefficients in…
This document develops general concepts useful for extracting knowledge embedded in large graphs or datasets that have pair-wise relationships, such as cause-effect-type relations. Almost no underlying assumptions are made, other than that…
In traditional topology optimization, the computing time required to iteratively update the material distribution within a design domain strongly depends on the complexity or size of the problem, limiting its application in real engineering…
In this paper, We develop the stratified de Rham theory on singular spaces using modern tools including derived geometry and stratified structures. This work unifies and extends the de Rham theory, Hodge theory, and deformation theory of…
We generalize the notion of a small sheaf of sets over a topological space or manifold to define the notion of a small stack of groupoids over an \'etale topological or differentiable stack. We then provide a construction analogous to the…
Following the classical results of Stong, we introduce a cohomological analogue of a core of a finite sheaved topological space and propose an algorithm for simplification in this category. In particular we generalize the notion of beat…
Localic and realizability toposes are two central classes of toposes in categorical logic, both arising through the Hyland-Johnstone-Pitts tripos-to-topos construction. We investigate their shared geometric features by providing an…
In this article, the theory of sheaves is studied from a categorical point of view. This perspective vastly generalizes the usual theory of sheaves of sets to a more abstract setting which allows us to investigate the theory of sheaves with…
We introduce a novel gradient descent algorithm extending the well-known Gradient Sampling methodology to the class of stratifiably smooth objective functions, which are defined as locally Lipschitz functions that are smooth on some regular…
We introduce an original notion of extra-fine sheaf on a topological space, and a variant (hyper-extra-fine) for which \v{C}ech cohomology in strictly positive degree vanishes. We provide a characterization of such sheaves when the…
Shape matching has been a long-studied problem for the computer graphics and vision community. The objective is to predict a dense correspondence between meshes that have a certain degree of deformation. Existing methods either consider the…
As data grows in size and complexity, finding frameworks which aid in interpretation and analysis has become critical. This is particularly true when data comes from complex systems where extensive structure is available, but must be drawn…
Predictive coding (PC) replaces global backpropagation with local optimization over weights and activations. We show that linear PC networks admit a natural formulation as cellular sheaves: the sheaf coboundary maps activations to edge-wise…
Recent work in Topological Deep Learning (TDL) seeks to generalize graph learning's preeminent $message \ passing$ paradigm to more complex relational structures: simplicial complexes, cell complexes, hypergraphs, and combinations thereof.…
Gerstenhaber and Schack ([GS]) developed a deformation theory of presheaves of algebras on small categories. We translate their cohomological description to sheaf cohomology. More precisely, we describe the deformation space of (admissible)…
We use sheaf theory and the six operations to define and study the (equivariant) homology of stacks. The construction makes sense in the algebraic, complex-analytic, or even topological categories.
For manifold learning, it is assumed that high-dimensional sample/data points are embedded on a low-dimensional manifold. Usually, distances among samples are computed to capture an underlying data structure. Here we propose a metric…
In this paper we study symmetric motion planning algorithms, i.e. such that the motion from one state A to another B, prescribed by the algorithm, is the time reverse of the motion from B to A. We experiment with several different notions…
In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves…
Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. Additionally,…