ӳ����ý

 
Warren Center for Network and Data Sciences
Complex Systems Group & Department of Mathematics
University of Pennsylvania
What can persistent homology see?

Abstract:  The usual framework for TDA takes as its starting point that a data set is sampled (noisily) from a manifold embedded in a high dimensional space, and provides a reconstruction of topological features of that manifold. However, the underlying algebraic topology can be applied to data in a much broader sense, carries much richer information about the system than just the barcodes, and can be fine-tuned so it sees only features of the data we want it to see. I will discuss this framework broadly, with focus on few of these alternative viewpoints, including applications to neuroscience and matrix factorization.

Functional Cancer Genomics
ӳ����ý of MIT and Harvard
Primer: What is persistent homology?

Abstract:  A fundamental question in big data analysis is if or how these points may be sampled, noisily, from an intrinsically low-dimensional geometric shape, called a manifold, embedded in a high dimensional “sensor” space. Topological data analysis (TDA) aims to measure the “intrinsic shape” of data and identify this manifold despite noise and the likely nonlinear embedding. I will discuss the basics of the fundamental tool in TDA called persistent homology, which assigns to a point cloud a count of topological features –roughly “holes” of various dimensions – with a measure of importance of each feature recorded in a “barcode” of the data to help distinguish the significant features from the noise.