相关论文: Persistence and Sheaves

Persistent homology computes the multiscale topology of a data set by using a sequence of discrete complexes. In this paper, we propose that persistent homology may be a useful tool for studying the structure of the landscape of string…

Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the…

A method is presented for the distributed computation of persistent homology, based on an extension of the generalized Mayer-Vietoris principle to filtered spaces. Cellular cosheaves and spectral sequences are used to compute global…

We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.

Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques.…

We describe an analogue of the notion of a perverse sheaf in the setting of the derived category of coherent sheaves on an algebraic stack. Under strong additional assumptions the construction of coherent "intersection cohomology" complexes…

In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution…

We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Cech theory and the theory of derived categories. That is to say, on…

We establish the foundations of the theory of persistent cohomology operations, derive decomposition formulas for wedge sums and products, and prove their Gromov-Hausdorff stability. We use these results to construct pairs of Riemannian…

Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that utilize persistence diagrams in statistical inference, a full Bayesian treatment remains…

We show that the cohomology table of any coherent sheaf on projective space is a convergent--but possibly infinite--sum of positive real multiples of the cohomology tables of what we call supernatural sheaves.

Computational topologists recently developed a method, called persistent homology to analyze data presented in terms of similarity or dissimilarity. Indeed, persistent homology studies the evolution of topological features in terms of a…

We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional…

Persistent homology (PH) studies the topology of data across multiple scales by building nested collections of topological spaces called filtrations, computing homology and returning an algebraic object that can be vizualised as a…

Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…

We study sheaves on holomorphic spaces of loops and apply this to the study of the complex, defined in \cite{BdSHK}, governing deformations of the \emph{Poisson vertex algebra} structure on the space of holomorphic loops into a Poisson…

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…

We find the sharp bounds on $h^0(F)$ for one-dimensional semistable sheaves $F$ on a projective variety $X$ by using the spectrum of semistable sheaves. The result generalizes the Clifford theorem. When $X$ is the projective plane…

This paper investigates the propreties of the persistence diagrams stemming from almost surely continuous random processes on $[0,t]$. We focus our study on two variables which together characterize the barcode : the number of points of the…

Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…