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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2020

    added this pointer:

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2020
    • (edited Oct 2nd 2020)

    added pointer to today’s

    diff, v9, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2022

    I have taken the liberty of re-writing the Idea-section from scratch.

    have also included a graphics meant to indicate the role of cycles and dually of cocycles

    have not yet included the kind of graphics that most intros to TDA are showing (essentially a stage in a Vietoris-Rips complex) Maybe later.

    diff, v15, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2022
    • (edited May 23rd 2022)

    I have expanded a little more, now there are two Idea-subsections:

    In the one on persistent homology/homotopy (here) I have highlighted that the implication of persistent cycles for actually interpreting data in practice is often unclear (and certainly not provided by the mathematics).

    In the next one on persistent cohomotopy (now here) I have made explicit that persistent Cohomotopy provides the answer to a concrete and common question about data sets.

    diff, v16, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2022

    have now produced (here) an animated gif which illustrates persistency of cycles.

    (Maybe the frame rate is somewhat too high? )

    diff, v18, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2022

    now uploaded an animated gif (here) which is meant to illustrate the idea of persistent cohomotopy

    diff, v19, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2022

    Under “Applications” I have added pointer to today’s

    • Daniel Spitz, Julian M. Urban, Jan M. Pawlowski, Confinement in non-Abelian lattice gauge theory via persistent homology [arXiv:2208.03955]

    using TDA for the recognition of instantons and confinement in lattice gauge theory. This looks interesting.

    diff, v26, current

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 12th 2022

    Is there anything important in this Quanta magazine article – After a Quantum Clobbering, One Approach Survives Unscathed?

    The idea seems to be that many claims that quantum algorithms outperform classical ones have been shown to be wrong, but not topological data analysis (TDA).

    So there’s

    • Lloyd, S., Garnerone, S. & Zanardi, P., Quantum algorithms for topological and geometric analysis of data. Nat Commun 7, 10138 (2016). doi:10.1038/ncomms10138

    These algorithms are believed to be classically intractable:

    • Casper Gyurik, Chris Cade, and Vedran Dunjko, Towards quantum advantage via topological data analysis, Quantum 6, 855 (2022). doi:10.22331/q-2022-11-10-855

    This claim is then considered through the lens of supersymmetry

    • Chris Cade, P. Marcos Crichigno, Complexity of Supersymmetric Systems and the Cohomology Problem (arXiv:2107.00011).

    But maybe this is all trumped by persistent cohomotopy as New Foundations for Topological Data Analysis.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2022
    • (edited Dec 12th 2022)

    The idea seems to be that many claims that quantum algorithms outperform classical ones have been shown to be wrong

    Depends on what you count as a “claim”. The field as such rests on theorems that quantum algorithms like Shor’s are provably more efficient than any classical counterpart.

    On a different but related note, I would be happy to hear of an example where traditional TDA (i.e. persistent homology of data spaces instead of Cohomotopy) solves an actual problem.

    Recently I had occasion to learn from the original author that the notion of persistent cycles was originally motivated from computer models of membrane proteins which had difficulty recognizing the prominent tubular “hole” going through these proteins (and hence through the cell membrane). When I asked whether, after inventing TDA from this motivation, it served to say anything novel about protein structure, the answer was plainly “No”. If I understood well.

    But inherently spatial data like protein structure would at least plausibly be an application for TDA. The real question is whether in generic real-world data (say in finance) the notion of persistent cycles has any practical meaning.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 12th 2022

    Just to be clear about the contrast, if any, in your view does the persistent cohomotopy of generic real-world data have any practical meaning?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2022
    • (edited Dec 12th 2022)

    Yes, that’s explained in New Foundations for TDA – Cohomotopy (schreiber). From the abstract:

    While a tool called persistent homology has become the signature method of TDA, it tends to produce answers that are either hard to interpret (persistent cycles) or impossible to compute (well groups). Both problems are solved by a variant method [FK17] which we may call persistent cohomotopy

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 12th 2022

    So I guess I’d need to know what “Find data meeting prescribed target with uncertainties” means in a real-world situation.

    Full of trigger words for why I chose the perilous path of an academic career over a safer financial one – “Portfolio Management in New Product Development”, “Risk-Reward bubble diagrams”.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2022
    • (edited Dec 12th 2022)

    This is the generic real-world situation, it seems to me: You want a design criterion x:Xx \colon X, you have a supply of d:Dd \colon D and a function s:D→Xs \colon D \to X. Now you want to see if within your available DD you can find dds with s(d)s(d) close to xx.

    In that finance example X=ℝ 2X = \mathbb{R}^2 is the risk/reward plane, DD is some collection of stocks or similar, x∈Xx \in X is your investment strategy, and now you ask persistent Cohomotopy to tell you whether you can succeed, to given approximation.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 12th 2022

    Ok, thanks. So that’s everywhere. Must be applications in machine learning.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2024

    added pointer to

    diff, v31, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2024

    and this one:

    diff, v31, current

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