Barcodes: The Persistent Topology of Data by Robert Ghrist.
Abstract:
This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets — persistent homology — and a novel representation of this algebraic characterization — barcodes. We sketch an application of these techniques to the classification of natural images.
From the article:
1. The shape of data
When a topologist is asked, “How do you visualize a four-dimensional object?” the appropriate response is a Socratic rejoinder: “How do you visualize a three-dimensional object?” We do not see in three spatial dimensions directly, but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended. We spend a significant portion of our first year of life learning how to infer three-dimensional spatial data from paired planar projections. Years of practice have tuned a remarkable ability to extract global structure from representations in a strictly lower dimension.
The inference of global structure occurs on much finer scales as well, with regards to converting discrete data into continuous images. Dot-matrix printers, scrolling LED tickers, televisions, and computer displays all communicate images via arrays of discrete points which are integrated into coherent, global objects. This also is a skill we have practiced from childhood. No adult does a dot-to-dot puzzle with anything approaching anticipation.
1.1. Topological data analysis.
Problems of data analysis share many features with these two fundamental integration tasks: (1) how does one infer high dimensional structure from low dimensional representations; and (2) how does one assemble discrete points into global structure.
Now are you interested?
Reminds me of another paper on homology and higher dimensions that I need to finish writing up. Probably not today but later this week.
Found thanks to: Christophe Lalanne’s A bag of tweets / Feb 2012.