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Isn't response surface methodology a form of topological data analysis? IIRC, it isn't used much since it has poor predictive power.
The company puts a lot of energy into making Gunnar's work accessible - maybe too much. Dig into his papers sometime.
Could you explain what you mean by "maybe too much"?
Just that the material featured most prominently on the site might be too imprecise for mathematicians and scientists with significant technical expertise. It would be nice if there was a convenient link I could pass along to informed, but highly skeptical, people with a genuine interest in understanding the platform on a deeper level.
The key with Ayasdi's work is that they manage to layer the TDA with different ML filters, which does stunningly well for the datasets they like talking about.

I also second the notion to read papers from the Carlsson lab, particularly one of the more application-oriented papers such as this one: http://www.nature.com/srep/2013/130207/srep01236/pdf/srep012...

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I'd love to see a success story of this type of analysis outside of their canned examples. I keep seeing them use the same datasets over and over again without any real benchmarks to state of the art. Its amazing how a data product is being sold without any empirical studies or benchmark datasets.

Ayasdi seems successful to me in that it has a lot of flash and their results make intuitive sense, but I don't understand how a practicing data scientist would use this.

I've seen it used "in the wild" in this publication: "Topographical transcriptome mapping of the mouse medial ganglionic eminence by spatially resolved RNA-seq" http://genomebiology.com/2014/15/10/486/abstract (pdf: http://linnarssonlab.org/pdf/Genome%20Biology%202014.pdf ) to cluster gene expression samples from mouse brain.

I'm working on a project where I'm using similar methods, but not from Ayasdi, to study cyclic phenomena in high-dimensional data.

Ok, so here it's used to cluster. There are tons of benchmark clustering datasets. Never seen it used on any of those.
The key paragraph "The projection is visualized as ... pictured as below." is very ambiguous, and completely missing the explanation of how the data was split into red, blue, and indigo clusters.