I have a PhD in math pure math, and I am fairly well-versed in algebraic topology and I've studied topological data analysis.
The idea is that you take your data points and expand them in space by putting circles around them. If there are features that are persistent over large variations of the sizes of the circles (or balls in higher-dimensional space), then those persistent features are said to be important reflections of the structure of your data.
The persistent features are characterized by homology, which is a tool that measures the basic topological shape of your data. Homology is actually extremely easy to calculate and doesn't require much advanced math. In fact, for data analysis purposes, you can focus purely on combinatorial aspects, in which case no advanced math is required.
In my opinion, it's a pretty idea and seems logical, but in practice it really seems to be limited in a few ways in terms of practical applications. For one, it often is weaker than more direct methods such as good experimental design and other cluster data methods that can be more directly used to tease out causal relationships.
For another, it seems most useful in the domain of comparing data changing over time or some other variable where the fundamental topological structure change in high-dimensional space in tidy ways. This isn't the case with a lot of real-world data.
In practial terms, you can think of this in the following way... there is only so much you can get out of data. And often, what you do need is not coarse topological features, which rarely have intrinsic meaning.
So my overall opinion of it is that while it might have some limited applications in highly specific fields, it will never become a general method that could be used in 99.9% of data science applications. (To be FAIR though, I also believe that a LARGE proportion of data science is also kind of useless too...data science has its fair share of snake oil.)
The idea is that you take your data points and expand them in space by putting circles around them. If there are features that are persistent over large variations of the sizes of the circles (or balls in higher-dimensional space), then those persistent features are said to be important reflections of the structure of your data.
The persistent features are characterized by homology, which is a tool that measures the basic topological shape of your data. Homology is actually extremely easy to calculate and doesn't require much advanced math. In fact, for data analysis purposes, you can focus purely on combinatorial aspects, in which case no advanced math is required.
In my opinion, it's a pretty idea and seems logical, but in practice it really seems to be limited in a few ways in terms of practical applications. For one, it often is weaker than more direct methods such as good experimental design and other cluster data methods that can be more directly used to tease out causal relationships.
For another, it seems most useful in the domain of comparing data changing over time or some other variable where the fundamental topological structure change in high-dimensional space in tidy ways. This isn't the case with a lot of real-world data.
In practial terms, you can think of this in the following way... there is only so much you can get out of data. And often, what you do need is not coarse topological features, which rarely have intrinsic meaning.
So my overall opinion of it is that while it might have some limited applications in highly specific fields, it will never become a general method that could be used in 99.9% of data science applications. (To be FAIR though, I also believe that a LARGE proportion of data science is also kind of useless too...data science has its fair share of snake oil.)