This is a really fascinating subject, and as of last year I had read through essentially all the major papers in the area, implemented the central algorithms, and given a few more-or-less casual talks on it.
There are two things that strike me as very different about topological data analysis. The first and primary thing is that it's not a silver bullet by any means. It's not like most machine learning and data mining, where you pick some parameters and wham-bam-thank-you-ma'am you have 80% accuracy. No, this kind of analysis gives you qualitative features of your data, and after all the topology is done there's still years of work before you arrive at a mathematical model that admits an algorithm that has accuracy comparable to the state of the art.
An interesting case study in this is Carlsson's work on texture classification. They ran their 3x3 image-patch database through their topological analysis algorithms and it said (essentially) "your data looks like a Klein bottle!" That sounds interesting and fun, but how can you actually use that to do anything? This followup paper [1] then gave an actual model of image patches as a Klein bottle, but even then there is still a ton of math to trudge through (specifically, functional analysis and differential geometry) before they actually got to an algorithm, and it still wasn't strictly better than the leading methods. The real benefit seems to be entirely in the science part of everything. They have a novel hypothesis for how the world acts which is demonstrably accurate. It's not like a support vector machine with well-chosen features where it's as useful as a black box in terms of understanding the world.
The second thing is that this field has legitimate and nontrivial results that come from category theory, eventually leading to algorithms or proofs that you can't hope for an algorithm. Carlsson has a nice survey for starers [2]. So everyone who loves to talk about category theory and programming on the internet now has a less abstract quiver of arrows (pun intended, if you're familiar with this research area).
Sensitivity analysis - http://en.wikipedia.org/wiki/Sensitivity_analysis is an interesting tradeoff between generating a specific algorithm and a block box, in that it tells you which parameters had the biggest impact on the output given some model like an SVM.
There are two things that strike me as very different about topological data analysis. The first and primary thing is that it's not a silver bullet by any means. It's not like most machine learning and data mining, where you pick some parameters and wham-bam-thank-you-ma'am you have 80% accuracy. No, this kind of analysis gives you qualitative features of your data, and after all the topology is done there's still years of work before you arrive at a mathematical model that admits an algorithm that has accuracy comparable to the state of the art.
An interesting case study in this is Carlsson's work on texture classification. They ran their 3x3 image-patch database through their topological analysis algorithms and it said (essentially) "your data looks like a Klein bottle!" That sounds interesting and fun, but how can you actually use that to do anything? This followup paper [1] then gave an actual model of image patches as a Klein bottle, but even then there is still a ton of math to trudge through (specifically, functional analysis and differential geometry) before they actually got to an algorithm, and it still wasn't strictly better than the leading methods. The real benefit seems to be entirely in the science part of everything. They have a novel hypothesis for how the world acts which is demonstrably accurate. It's not like a support vector machine with well-chosen features where it's as useful as a black box in terms of understanding the world.
The second thing is that this field has legitimate and nontrivial results that come from category theory, eventually leading to algorithms or proofs that you can't hope for an algorithm. Carlsson has a nice survey for starers [2]. So everyone who loves to talk about category theory and programming on the internet now has a less abstract quiver of arrows (pun intended, if you're familiar with this research area).
[1]: http://comptop.stanford.edu/u/preprints/KleinBottleTextureAn... [2]: http://www.ayasdi.com/_downloads/Topology_and_Data.pdf