7 ms·
Algebraic Topology for Data Scientists
- KhoomeiK 3y agoI 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 agoI 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 agoThis 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 agoor category theory phase
- wenc 3y agoOr 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 agoThere 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 agoI’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 agoI 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 agoWhen 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 agoMy 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 agoWould love to hear more.
- KhoomeiK 3y agoPlease 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.
- xiaodai 3y agowhat does this have to do with data science? Does this guy even know any data science?
- pizza 3y agoAlgebraic topology is a mathematical toolkit for analyzing connectedness/structure. Pretty applicable to data science.
- wenc 3y ago> Pretty applicable to data science. Could you expand on that? Maybe I'm missing something but I browsed the PDF and am having trouble finding anything that would be deployed to a production system or produced as part of an analysis in a data science workflow. I think there some very niche applications in manifold learning (e.g. UMAP) that is somewhat useful, but in my decade in this space, people have tried to find topology applications but ultimately we end up going back to the basic tools of statistics and machine learning.
- moravak1984 3y agoWell, you are not alone. As a Math PhD, I want to believe that there is something less trivial than glorified linear regression to this whole AI/ML mess, but well, there isn't... It's non-linear transforms + holy Mary's all the way.
- wilseypa 3y agothere are a decent set of publications describing data analysis with tda/ph; a source with citations to several can be found in: Otter, N., Porter, M.A., Tillmann, U. et al. A roadmap for the computation of persistent homology. EPJ Data Sci. 6, 17 (2017). https://doi.org/10.1140/epjds/s13688-017-0109-5 https://doi.org/10.1140/epjds/s13688-017-0109-5
- pizza 3y agoIf you want to optimize parameters to maximize a function, calculus is a natural process for that. If you want to treat data as having richer structure than just a point cloud in a metric space, topology is a natural process for that. Network characterization (eg community/bottleneck/robustness measurement) is not that niche imo, but I guess data science defaults to meaning calculus because calculus is fast and cheap to compute compared to topological methods. There are classes of answers that I don’t see how you could find them without resorting to something like TDA. Stuff like characterizing the Betti numbers of neuronal circuits to profile signal routing redundancy. The economics of applying TDA at scale (at least at one that does not require eg nation-state level compute) don’t work well atm I think: small problems will quickly kill the beefiest cpu/ram combo you can find if you try to do basic persistent homology on them, so either you scale your compute like crazy for results that probably won’t give you jackpot margins wrt competitors, or just do whatever is cheap and quick and good enough. A remark: I think tda is in an era not too unlike deep learning after the proliferation of backprop and little deep nets, but before backprop-on-gpus. People said all kinds of things about how deep learning was a gimmick, was too impractical to apply to real world problems, etc. I remain curious either way.
- pietroppeter 3y agoThis seems to be mostly focused on math, what could be a good TDA resource more focused on applications?
- pietroppeter 3y agoLooking in past HN the most relevant results seems this from 8 years ago linking to a company that does not seem to exist anymore: https://news.ycombinator.com/item?id=9487872 https://news.ycombinator.com/item?id=9487872
- pietroppeter 3y agoMmh no seems they went through at least a couple of renamings: https://www.netreveal.ai/resources/blogs/ayasdis-next-chapter/ https://www.netreveal.ai/resources/blogs/ayasdis-next-chapte...
- peepeepoopoo30 3y ago[flagged]
- emmanueloga_ 3y agoNOTE: TDA = Topological data analysis [0]. See also: 3LA [1] (sigh). -- 0: https://en.wikipedia.org/wiki/Topological_data_analysis https://en.wikipedia.org/wiki/Topological_data_analysis. 1: https://en.wikipedia.org/w/index.php?title=3LA&redirect=no https://en.wikipedia.org/w/index.php?title=3LA&redirect=no
- deleted 3y ago[deleted]
- srvmshr 3y agoFor those interested, there is a really nice course with lecture notes taught by Vidit Nanda (Oxford). https://people.maths.ox.ac.uk/nanda/cat/ https://people.maths.ox.ac.uk/nanda/cat/ Unlike what people commented elsewhere, TDA has growing application in computer graphics, 3D reconstructions and computer vision. PS: I am told the new version of this course is narrated/illustrated by Robert Ghrist. He's famous for his foundational calculus course
- northzen 3y agoCan you give some hints or examples of TDA in 3D reconstructions and computer vision? I'm interested in this topic, espectially exploring it from topological view.
- viknesh 3y agoI took some algebraic topology in school and I think it's a fascinating subject. As some other commenters mentioned, though, it was difficult to try to skim this book to find high level applications. For anybody else that's interested, I think there are some applications in chapters 5-7.
- pid-1 3y agoThis is a maths book with zero Data Science content. If you're wondering how Algebraic Topology can be used to solve real world problems, you will still wonder 300 pages later.
- usgroup 3y agoLooking through the contents I thought the same thing. Seems like a lot of theory with only a handful of attached methods. I think one might be better of just reading about uMAP, Mapper and persistent homology directly . At least the last two are very simple and don’t require advanced maths.
- bigbillheck 3y agoIt's "Algebraic Topology for Data Scientists", not "Even More Data Science with some Algebraic Topology Included for Data Scientists".
- hgsgm 3y ago"This book gives a thorough introduction to topological data analysis (TDA), the application of algebraic topology to data science."
- genoveffo 3y agoIs there a clear application to a practical, real world data set where TDA is the state of the art? Don’t get me wrong, I love maths and topology in particular but I’m trying to understand what are the advantages that these techniques bring to the table
- causalmodels 3y agoI’ve used algebraic topology for certain robotics tasks when statistical saftey wasn’t good enough and when trying to generalize techniques to learn on graphs. The results are spectacular, but the application set is indeed very limited.
- deepcorner123 3y agoany references you can share? would love to read more about robotics applications of these approaches
- causalmodels 3y agoThis is a decent place to get started https://arxiv.org/pdf/math/0009118.pdf https://arxiv.org/pdf/math/0009118.pdf You can extend some of the thinking in this paper to cover action spaces for deep RL models to really have some fun!
- necroforest 3y agono, at least not for anything non-contrived.
- renatyv 3y agoDon't get me wrong, I like math and had fun skimming through the book. But 4 pure math chapters to get to "growing circles"? Really? It feels like these algorithms can be explained using much simpler terms.
- vouaobrasil 3y agoI 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.)
- jebarker 3y agoI also have a PhD in Algebraic Topology and feel similarly. One application I'd like to see explored more is the use of differentiable topological invariants as loss functions for training neural nets. Loss functions to measure topological similarity don't really exist in practice yet and could be very useful.
- maddimini 3y agoWhat do you mean by "other cluster data methods that can be more directly used to tease out causal relationships"? What clustering approaches can tease out causal relationships? Thanks in advance!
- srean 3y agoThere have been many comments that have pointed out the poverty of real ML applications that ride on algebraic topology. The problem is this -- topological spaces on their own have a lot less structure than say a vector space or a metric space. Most of the real world data that ML applications deal with today have more structure and are thus handled using vector space methods or metric space methods or manifold methods. It is also the right thing to do. When you have structure you shoukd use it. TDA would be useful for data sets where the members have no clear analogue of a distance, or the distances cannot be trusted, or where vector embeddings do not make sense.
- outlace 3y agoIf you're interested in algebraic topology to do topological data analysis, I wrote a series of blog posts about it that I believe is quite accessible even to those with limited math backgrounds: http://outlace.com/TDApart1.html http://outlace.com/TDApart1.html
- ConnorMooneyhan 3y ago> most mathematicians have never been exposed to it I'm not sure what graduate math programs don't have Algebraic Topology. It's pretty fundamental.
- bannedbybros 3y ago[dead]
- bigbillheck 3y agoMy math program didn't require graduate topology for most students, they offered a topology course but most of it was general, not algebraic.
- godelski 3y ago322 pages and no links or bookmarks! Come on! Instructions to add: - Click other formats - Download source - Add this to the includes ``` \usepackage[bookmarks,linktocpage=true]{hyperref} \hypersetup{ colorlinks, linktoc=all, linkcolor={blue}, } \makeindex ``` - recompile Presto, you got the same document except now we can click the links in the table of contents, the citations, and there's a bookmark section on the side that allows us to navigate the document instead of just scrolling.