• Information - Data - Data management - Data analysis - Topological data analysis
• ont-uid: iejei1th
• This is the main entry for "topological data analysis".
• In applied mathematics, topological based 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.
The initial motivation is to study the shape of data.
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.
Moreover, its mathematical foundation is also of theoretical importance.
The unique features of TDA make it a promising bridge between topology and geometry.
• https://en.wikipedia.org/wiki/Topological_data_analysis
• https://en.wikipedia.org/wiki/Category:Computational_topology
• https://en.wikipedia.org/wiki/Category:Data_analysis
• https://en.wikipedia.org/wiki/Category:Homology_theory
• https://en.wikipedia.org/wiki/Category:Applied_mathematics
• See also:
(2022-12-07, https://www.quantamagazine.org/after-a-classical-clobbering-a-quantum-advantage-remains-20221207/) "After a Quantum Clobbering, One Approach Survives Unscathed.
A quantum approach to data analysis that relies on the study of shapes will likely remain an example of a quantum advantage - albeit for increasingly unlikely scenarios." Quantum computers get a lot of hype, but the truth is we're still not sure what they'll be good for.
These devices leverage the peculiar physics of the subatomic world and have the potential to perform calculations that regular, classical computers simply can't.
But it's proved difficult to find examples of any algorithms with a clear "quantum advantage" that enables performance beyond the reach of classical machines.
For most of the 2010s, many computer scientists felt one particular group of applications had a great shot at finding this advantage.
Certain data-analysis calculations would be exponentially faster when they were crunched by a quantum computer.
Then along came Ewin Tang.
As an 18-year-old recent college grad in 2018, she found a new way for classical computers to solve these problems, smacking down the advantage the quantum algorithms had promised.
For many who work on quantum computers, Tang's work was a reckoning.
"One by one, these super exciting use cases just got killed off," said Chris Cade, a theoretical computer scientist at the Dutch quantum computing research center QuSoft.
But one algorithm survived unscathed: a quantum twist on a niche mathematical approach for studying the "shape" of data, called topological data analysis (TDA).
After a flurry of papers in 2022-09, researchers now believe that these TDA calculations lie beyond the grasp of classical computers, perhaps due to a hidden connection to quantum physics.
But this quantum advantage may only occur under highly specific conditions, calling its practicality into question.
Seth Lloyd, a quantum mechanical engineer at the Massachusetts Institute of Technology who co-created the quantum TDA algorithm, remembers its origin vividly.
He and fellow physicist Paolo Zanardi were attending a quantum physics workshop in an idyllic town in the Pyrenees mountains in 2015.
A few days into the conference, they were skipping talks to hang out on the hotel patio as they tried to wrap their heads around a "crazy abstract" mathematical technique they had heard about for analyzing data.
Zanardi had fallen in love with the math underlying TDA, which was rooted in topology, a branch of mathematics concerned with features that remain when shapes are squashed, stretched or twisted.
"This is one of those branches of mathematics which just percolates everything," said Vedran Dunjko, a quantum computing researcher at Leiden University.
"It's everywhere." One of the field's central questions is the number of holes in an object, called a Betti number.
Topology can extend beyond our familiar three dimensions, allowing researchers to calculate the Betti numbers in four-, 10- and even 100-dimensional objects.
This makes topology an appealing tool for analyzing the shapes of big data sets, which can also include hundreds of dimensions of correlations and connections.
Currently, classical computers can only calculate Betti numbers up to around four dimensions.
On that Pyrenean hotel patio, Lloyd and Zanardi attempted to break that barrier.
After about a week of discussion and scribbled equations, they had the bare bones of a quantum algorithm that could estimate the Betti numbers in data sets of very high dimensions.
They published it in 2016, and researchers welcomed it into the group of quantum applications for data analysis that they believed had a meaningful quantum advantage.
Within two years, TDA was the only one that hadn't been impacted by Tang's work.
While Tang admits that TDA is "genuinely different from the others," she and other researchers were left to wonder to what degree its escape might have been a fluke. ... Crichigno suspects TDA's resilience points to an inherent - and wholly unexpected - connection to quantum mechanics.
This link comes from supersymmetry, a theory in particle physics that proposes a deep symmetry between the particles that make up matter and those that carry forces.
It turns out, as the physicist Ed Witten explained in the 1980s, that the mathematical tools of topology can easily describe these supersymmetric systems.
Inspired by Witten's work, Crichigno has been inverting this connection by using supersymmetry to study topology.
"That's nuts.
That's a really, really, really strange connection," said Dunjko, who was not involved in Crichigno's work.
"I get goose bumps.
Literally." This hidden quantum connection might be what set TDA apart from the rest, said Cade, who has worked with Crichigno on this.
"This really is, in essence, a quantum mechanical problem, even though it doesn't look like it," he said. ...
• (2022-11-10, https://quantum-journal.org/papers/q-2022-11-10-855/) "Towards quantum advantage via topological data analysis."