27 High - Dimensional Topological Data Analysis
نویسنده
چکیده
simplicial complex: Given a set X, an abstract simplicial complex C with vertex set X is a set of finite subsets of X, the simplices, such that the elements of X belong to C and if σ ∈ C and τ ⊂ σ then τ ∈ C. Homology: Intuitively, homology (with coefficient in a field) associates to any topological space X, a family of vector spaces, the so-called homology groups Hk(X), k = 0, 1, . . ., each of them encoding k-dimensional topological features of X. A fundamental property of homology is that any continuous function f : X → Y induces a linear map f∗ : Hk(X) → Hk(Y ) between homology groups that encodes the way the topological features of X are mapped to the
منابع مشابه
Methods for regression analysis in high-dimensional data
By evolving science, knowledge and technology, new and precise methods for measuring, collecting and recording information have been innovated, which have resulted in the appearance and development of high-dimensional data. The high-dimensional data set, i.e., a data set in which the number of explanatory variables is much larger than the number of observations, cannot be easily analyzed by ...
متن کاملTopological Relationship Between One-Dimensional Box Model and Randić Indices in Linear Simple Conjugated Polyenes
The alternative double bonds and conjugation in the polyene compounds are one of the main properties in these compounds. Each carbon-carbon bonds in a polyene compound along the chain has appreciable double-bond character. The p-electrons are therefore not localized but are relatively free to move throughout the entire carbon skeleton as an one-dimensional box. The skeleton be considered as a r...
متن کاملTopological Data Analysis
Scientific data is often in the form of a finite set of noisy points, sampled from an unknown space, and embedded in a high-dimensional space. Topological data analysis focuses on recovering the topology of the sampled space. In this chapter, we look at methods for constructing combinatorial representations of point sets, as well as theories and algorithms for effective computation of robust to...
متن کاملExtracting Knowledge from the Geometric Shape of Social Network Data Using Topological Data Analysis
Topological data analysis is a noble approach to extract meaningful information from high-dimensional data and is robust to noise. It is based on topology, which aims to study the geometric shape of data. In order to apply topological data analysis, an algorithm called mapper is adopted. The output from mapper is a simplicial complex that represents a set of connected clusters of data points. I...
متن کاملStatistical Inference for Topological Data Analysis PhD Thesis Proposal
Topological Data Analysis (TDA) is an emerging area of research at the intersection of algebraic topology and computational geometry, aimed at describing, summarizing and analyzing possibly high-dimensional data using low-dimensional algebraic representations. Recent advances in computational topology have made it possible to actually compute topological invariants from data. These novel types ...
متن کامل