I'm a data scientist and mathematician at Ayasdi.

For a variety of reasons we don't participate in competitive data science - however - we do a lot of actual science that you can use to get some idea on the power of the technique:

http://www.ayasdi.com/resources/#publications

Work using related techniques but not directly Ayasdi's implementations of TDA can be found here (although for some reason the page hasn't been updated in the last year):

http://appliedtopology.org/preprints/

Although you'll have to take my word for it - we are frequently brought in when people have tried the standard techniques and have run out of ideas. In those cases we have to show that we can improve on those models. We do that by discovering how some outcome of interest is spread (localized) in the data. You can think of this as finding different local descriptions of a phenomena. Many methods will fine the dominant description but miss more subtle variations - we find a more complete picture.

For example, if you're looking at fraud (credit card, login etc) velocity-based measures may catch machine-assisted fraud but miss manual and more sophisticated fraud. Different kinds of fraud have different (local) descriptions.

From a data scientist workflow this means that you (1) select features. (2) create topological summaries.(3) find regions where fraud is localized (4) find underlying reasons why the fraud is being localized in these groups (5) use this information to build better models (exactly how you use this information is another long post).

It's important to note that I didn't have to think of the reasons for all of the different types of fraud ahead of time - the groupings and shape summary guide my feature search (to be fair this is usually an iterative process initial features --> shape localizations --> more/different features etc.).

I ansered a similar question as part of a kdnuggets interview. I'll excerpt it here:

Q3. What are the unique benefits of Topological Data Analysis (TDA) over other approaches?

Topology is the study and description of shape. In Big Data problems, shape arises because you have a notion of similarity or distance between data points. This can be something like Euclidean distance, correlations, a weighted graph distance or even something more esoteric. Shape is exploited in machine learning by bringing in some additional information such as “my data has well defined clusters or classes”; “this outcome is linear “; or “my signal is periodic”. Then you would use specialized tools to apply models based on this information.

Topology adds the ability to understand and describe the shape without imposing additional model information, which can be biased and misleading. This leads to a number of concrete benefits, such as predictive model improvement and a better understanding of your data.

This seems like a trivial point, but can be key to solving complex problems with a high degree of accuracy. A simple example of this comes from hospital predictive models. Hospitals want to measure how sick people are and collect a variety of clinical information (blood pressure, heart rate, temperature, breathing rate, oxygen levels etc.) or genetic information (gene expression levels). Typically, they fit a linear regression model that predicts how “sick” patients are. The underlying assumption is that there is a near linear relationship between symptoms and “sickness”.

One of Ayasdi’s academic collaborators took gene expression data for people at different stages of malaria. When examined using TDA, he found patients all lying on a circle sitting inside of a high dimensional space (~1000 features). While in retrospect the circle is obvious, your path from being healthy to sickness and back to healthy does not track up and down through the same set of symptoms, and yet nobody had thought to look for the circle.

Most real world data sets that I look at are larger and more complicated than this example and we find a variety of structures — cluster, flares, loops and higher dimensional structures — all appearing in a single data set. It is nearly impossible to guess or hypothesize the right structures ahead of time, and TDA is a tool to understand your data in an unbiased way, revealing its true complexity.

The complete interview is here:

http://www.ayasdi.com/blog/topology/kdnuggets-interview-anth...