4 ms·

I feel like everyone in data science or ML that has some higher math background goes through their TDA phase, trying to apply persistent homology to everything.

I feel like everyone in data science or ML that has some higher math background goes through their TDA phase, trying to apply persistent homology to everything. My view now is that the applications are quite limited, but the math is still cool.

jsunderland323 3y ago
I think there’s some pretty cool applications to be built upon homotopy type theory/ univalent foundations. One use case I’m working on is building a structured version control system that allows versioned documents to be translated to code at compile time. Think rich text translations or other static assets that you may want to change at runtime. One really interesting exploitation of the univalence axiom is that you can make hot updates to compiled code without having to recompile the entire system by knowing if the identity paths of the updated document state is a topological subset of the document used at compile time. If it is, you know the updated generated code is isomorphic with the compile time code, meaning it’s safe to swap out. That’s pretty mind blowing and (seems very) practical to me. Basically allows for higher order configuration editing.
- mathisfun123 3y ago
  >I think there’s some pretty cool applications to be built upon homotopy type theory/ univalent foundations. constructive type theory and TDA are very different things even if they do both have `topo` in their names.
tnecniv 3y ago
This has generally been my experience with a few more abstract areas of math. Maybe it’s just me and my domain of interest, but I can never seem to find an application that justifies the effort required to learn it. I’m glad some people are working on it, because maybe that application will show up tomorrow and I find it conceptually neat, but there’s only so much time in the day.
latenightcoding 3y ago
or category theory phase
wenc 3y ago
Or Bayesian Data Analysis phase. This is a bit unfair since Bayesian techniques are useful for when you want to reason about limited data. But as soon as you have a fair sized dataset, frequentist techniques are typically computationally simpler and faster (no need for MCMC) and scale much better.
- muraiki 3y ago
  There are other benefits to Bayesian data analysis besides being able to handle limited data. There are problems with the outputs of frequentist analysis around the quantification of uncertainty. For instance, from simulation studies we know that the aleatoric coverage probability for confidence intervals of a selected confidence level varies depending on the size of the difference in plausibility between the null and alternative hypotheses. And a given confidence interval says nothing about epistemic uncertainty for this particular experiment. This can make the outputs of frequentist analysis difficult for stakeholders to utilize, whereas Bayesian epistemic probabilities are generally more easily understand by stakeholders, and can directly feed quantitative decision analysis methods. A good introduction to some additional problems with frequentist methods vs Bayesian and likelihoodist methods is this: https://gandenberger.org/2014/08/26/intro-to-statistical-methods-3/ https://gandenberger.org/2014/08/26/intro-to-statistical-met... An interesting book on adapting frequentist methods to create confidence distributions that can better express uncertainty and can optionally incorporate prior information using likelihood functions is this: https://www.cambridge.org/core/books/confidence-likelihood-probability/143A34F11FB3D6F611F78E27C6D2CA5A#fndtn-information https://www.cambridge.org/core/books/confidence-likelihood-p...
  - wenc 3y ago
    I’ve found that in many applications, the difference between a frequentist analysis and a Bayesian one is unlikely to make a difference in the decision making (even with UQ). I’m sure there are fields where such statistical rigor is called for (where the quality of the data is so high and accurate that the variation is in the analysis — often the case with machine data). For everything else there’s so much error. Being to quantify uncertainty is great — it’s a signal we need to collect more and better data. But so often we have to move ahead with uncertain data. Interestingly, in business, taking action (even if wrong) produces outcomes that are much better signals to learn from than having statistically rigorous analyses, so many times there’s a bias for action rather than obsession over analysis. But of course in some fields being wrong is costly (like clinical trials) so I can see UQ being more useful and prominent there.
ninepoints 3y ago
I forget who it was, but I think it was Ghrist that mentioned in his TDA book that expectations were to be extremely tempered before trying to apply TDA to a given problem domain.
loubbrad 3y ago
When I was studying Mathematics I initially thought TDA was magic. I remember seeing MATLAB correctly compute the Homology groups of a Torus from a randomly sampled point cloud. As I veer farther into the CS/ML world I've come to appreciate that most of the advertised practical applications of higher Mathematics are quite niche. Having said that, Algebraic Topology is still one of my favorite areas of Mathematics from a purely mathematical viewpoint.
causalmodels 3y ago
My undergrad research was on CW complexes and I did spend the following decade trying find applications for it in ML. It has worked out approximately twice, but my god is it powerful when it naturally fits a problem space.
- srean 3y ago
  Would love to hear more.
- KhoomeiK 3y ago
  Please do elaborate! This [1] is pretty much the only succesful use case I've seen so far. The use case in NLP [2] I've seen was a negative result. [1] https://arxiv.org/abs/1802.04443 https://arxiv.org/abs/1802.04443 [2] https://arxiv.org/abs/1705.10900 https://arxiv.org/abs/1705.10900
deleted 3y ago
[deleted]
slavapestov 3y ago
> My view now is that the applications are quite limited, but the math is still cool. This is true of everything, of course. Everyone knows that “practical applications” is just a flimsy excuse to get funding for your cool math project. I’ve found that the main benefit of learning math is not the practical applications of the theorems and algorithms themselves, but just the practice you get from using this approach to thinking. This also frees you up to learn the math you find interesting and not what you think might be immediately useful in your day job.

jsunderland323 3y ago

I think there’s some pretty cool applications to be built upon homotopy type theory/ univalent foundations. One use case I’m working on is building a structured version control system that allows versioned documents to be translated to code at compile time. Think rich text translations or other static assets that you may want to change at runtime. One really interesting exploitation of the univalence axiom is that you can make hot updates to compiled code without having to recompile the entire system by knowing if the identity paths of the updated document state is a topological subset of the document used at compile time. If it is, you know the updated generated code is isomorphic with the compile time code, meaning it’s safe to swap out. That’s pretty mind blowing and (seems very) practical to me. Basically allows for higher order configuration editing.

mathisfun123 3y ago
>I think there’s some pretty cool applications to be built upon homotopy type theory/ univalent foundations. constructive type theory and TDA are very different things even if they do both have `topo` in their names.

tnecniv 3y ago

This has generally been my experience with a few more abstract areas of math. Maybe it’s just me and my domain of interest, but I can never seem to find an application that justifies the effort required to learn it. I’m glad some people are working on it, because maybe that application will show up tomorrow and I find it conceptually neat, but there’s only so much time in the day.

latenightcoding 3y ago

or category theory phase

wenc 3y ago

Or Bayesian Data Analysis phase. This is a bit unfair since Bayesian techniques are useful for when you want to reason about limited data. But as soon as you have a fair sized dataset, frequentist techniques are typically computationally simpler and faster (no need for MCMC) and scale much better.

muraiki 3y ago
There are other benefits to Bayesian data analysis besides being able to handle limited data. There are problems with the outputs of frequentist analysis around the quantification of uncertainty. For instance, from simulation studies we know that the aleatoric coverage probability for confidence intervals of a selected confidence level varies depending on the size of the difference in plausibility between the null and alternative hypotheses. And a given confidence interval says nothing about epistemic uncertainty for this particular experiment. This can make the outputs of frequentist analysis difficult for stakeholders to utilize, whereas Bayesian epistemic probabilities are generally more easily understand by stakeholders, and can directly feed quantitative decision analysis methods. A good introduction to some additional problems with frequentist methods vs Bayesian and likelihoodist methods is this: https://gandenberger.org/2014/08/26/intro-to-statistical-methods-3/ https://gandenberger.org/2014/08/26/intro-to-statistical-met... An interesting book on adapting frequentist methods to create confidence distributions that can better express uncertainty and can optionally incorporate prior information using likelihood functions is this: https://www.cambridge.org/core/books/confidence-likelihood-probability/143A34F11FB3D6F611F78E27C6D2CA5A#fndtn-information https://www.cambridge.org/core/books/confidence-likelihood-p...
- wenc 3y ago
  I’ve found that in many applications, the difference between a frequentist analysis and a Bayesian one is unlikely to make a difference in the decision making (even with UQ). I’m sure there are fields where such statistical rigor is called for (where the quality of the data is so high and accurate that the variation is in the analysis — often the case with machine data). For everything else there’s so much error. Being to quantify uncertainty is great — it’s a signal we need to collect more and better data. But so often we have to move ahead with uncertain data. Interestingly, in business, taking action (even if wrong) produces outcomes that are much better signals to learn from than having statistically rigorous analyses, so many times there’s a bias for action rather than obsession over analysis. But of course in some fields being wrong is costly (like clinical trials) so I can see UQ being more useful and prominent there.

ninepoints 3y ago

I forget who it was, but I think it was Ghrist that mentioned in his TDA book that expectations were to be extremely tempered before trying to apply TDA to a given problem domain.

loubbrad 3y ago

When I was studying Mathematics I initially thought TDA was magic. I remember seeing MATLAB correctly compute the Homology groups of a Torus from a randomly sampled point cloud. As I veer farther into the CS/ML world I've come to appreciate that most of the advertised practical applications of higher Mathematics are quite niche. Having said that, Algebraic Topology is still one of my favorite areas of Mathematics from a purely mathematical viewpoint.

causalmodels 3y ago

My undergrad research was on CW complexes and I did spend the following decade trying find applications for it in ML. It has worked out approximately twice, but my god is it powerful when it naturally fits a problem space.

srean 3y ago
Would love to hear more.
KhoomeiK 3y ago
Please do elaborate! This [1] is pretty much the only succesful use case I've seen so far. The use case in NLP [2] I've seen was a negative result. [1] https://arxiv.org/abs/1802.04443 https://arxiv.org/abs/1802.04443 [2] https://arxiv.org/abs/1705.10900 https://arxiv.org/abs/1705.10900

deleted 3y ago

[deleted]

slavapestov 3y ago

> My view now is that the applications are quite limited, but the math is still cool. This is true of everything, of course. Everyone knows that “practical applications” is just a flimsy excuse to get funding for your cool math project. I’ve found that the main benefit of learning math is not the practical applications of the theorems and algorithms themselves, but just the practice you get from using this approach to thinking. This also frees you up to learn the math you find interesting and not what you think might be immediately useful in your day job.