Topological Computing

Topology underpins geometry and a fascinating topic on its own. When speaking of topological quantum computing it emphasizes the resistance to noise-induced errors and is related to the topology of knots and braids. In the context of machine learning and AI Topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools. TDA has combined algebraic topology and other tools from pure mathematics to allow mathematically rigorous study of “shape”. The main tool is persistent homology, an adaptation of homology to point cloud data. Persistent homology has been applied to many types of data across many fields.

Topological Computing

Topology and business

What is persistent homology?

Stochastic Neighbor Embedding on MNIST

Topological Data Analysis

Discrete calculus on graphs