[HN Gopher] An invariant from category theory solves a problem i...
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An invariant from category theory solves a problem in mathematical
ecology [pdf]
Author : peanutcrisis
Score : 83 points
Date : 2023-07-23 15:39 UTC (7 hours ago)
(HTM) web link (www.maths.ed.ac.uk)
(TXT) w3m dump (www.maths.ed.ac.uk)
| LudwigNagasena wrote:
| An interesting result, but saying that "the maximum diversity
| problem is completely solved by an invariant that comes from
| category theory" sounds like a parody. In the end it is just a
| mathematical result with a biological metaphor. But who knows,
| maybe in near future it will be used for solving diversity
| problems of boards of directors.
| jmount wrote:
| I am actually sympathetic to category theory, and the research at
| hand. That being said, what was presented might be interesting if
| left in terms of linear algebra or graph theory (both legitimate
| fields). Trying to put a category theory gloss on it just makes
| it look hollow.
| catgary wrote:
| It's literally the result of a category theorist "following his
| nose" and solving a problem - mathematical ecologists are
| perfectly capable of doing some graph theory/linear algebra!
| ognyankulev wrote:
| The book that includes the results from these slides has broader
| scope, and also can be downloaded for free from arXiv:
| https://www.maths.ed.ac.uk/~tl/ed/
|
| "The starting point is the connection between diversity and
| entropy. We will discover:
|
| * how Shannon entropy, originally defined for communications
| engineering, can also be understood through biological diversity
| (Chapter 2);
|
| * how deformations of Shannon entropy express a spectrum of
| viewpoints on the meaning of biodiversity (Chapter 4);
|
| * how these deformations provably provide the only reasonable
| abundance-based measures of diversity (Chapter 7);
|
| * how to derive such results from characterization theorems for
| the power means, of which we prove several, some new (Chapters 5
| and 9).
|
| Complementing the classical techniques of these proofs is a
| large-scale categorical programme, which has produced both new
| mathematics and new measures of diversity now used in scientific
| applications. For example, we will find: [...]"
|
| "The question of how to quantify diversity is far more
| mathematically profound than is generally appreciated. This book
| makes the case that the theory of diversity measurement is
| fertile soil for new mathematics, just as much as the
| neighbouring but far more thoroughly worked field of information
| theory"
| sesm wrote:
| Why not define ecological diversity as number of distinct
| biological species living in the area?
| autopoiesis wrote:
| This is precisely the question answered by the OP. The answer
| is, "because there is a whole spectrum of things you might
| mean by 'diversity', of which 'number of distinct species' is
| only one extremum".
| Scarblac wrote:
| And also, I assume, because the concept of "species" isn't
| all that well defined?
| sesm wrote:
| It is well defined: a group of living organisms
| consisting of similar individuals capable of exchanging
| genes or interbreeding
| sdenton4 wrote:
| I invite you to examine the notes of the international
| ornithological congress... The difference between species
| and subspecies is quite subtle, and subject to
| interpretation, because no one is really going to do the
| experiment to find out if two individuals of
| geographically district populations can actually still
| interbreed.
| LudwigNagasena wrote:
| Ring species make your definition non-transitive. The
| same with species that can interbreed but exhibit hybrid
| breakdown.
| Scarblac wrote:
| So if you have a few grams of soil and want to know how
| many species of micro organisms are in there, you're
| setting them up with dates to see which ones will end up
| breeding?
| sesm wrote:
| Does he provide an example of other definition of diversity
| that makes sense in biological context?
| MostlyStable wrote:
| Yes, the two extremes are captured by the common metrics
| of "species richness" which is the pure "how many unique
| species are there", and "species evenness", which depends
| on how evenly distributed the species are. A community in
| which 99% of individuals are species A and the remaining
| 1% are from species B-G is exactly as species rich as a
| community in which there are equal numbers of individuals
| of each species, but it is much less even (and therefore,
| under one extreme of diversity, less diverse). In
| different contexts and for different ecological
| questions, these two different versions of diversity can
| matter more or less, and there are metrics which take
| both into account, but this is a fully generalized
| solution which shows you relative diversity along the
| entire spectrum from "all I care about is richness" to
| "all I care about is evenness".
|
| -edit- by the way, since it may not be obvious to
| everyone, the reason why an ecologist might care bout
| evenness is because extremely rare species are often not
| very important to the wider community. From an ecological
| function perspective, there is very little difference
| between my above example of the 99%/1% community and a
| community that is 100% species A. So an community with
| two, equally populous species might have more functional
| diversity than a community with one very abundant species
| and several more, very rare species.
| contravariant wrote:
| Ah I was wondering about that. Their formula look suspiciously
| like the definition of Renyi entropy.
|
| I'm not too sure where the category theoretical stuff enters
| though. They mention that metric spaces have a magnitude, but
| their end result looks more like a channel capacity (with the
| confusion matrix being the probability to confuse one species
| with another). Which, you know, makes sense, if you've got 'N'
| signals but they're so easily confused with one another that
| you can only send 'n' signals worth of data then your channels
| are not too diverse.
|
| They do mention that this is equivalent to some modified
| version of the category theoretical heuristic, but is that
| really interesting? The link to Euler characteristic is
| intriguing, but from the way they end up at their final
| definition I'm not sure if _metric_ spaces are really the
| natural context to talk about these things. It almost feels
| like they 've stepped over an enriched category that would
| provide a more natural fit.
| autopoiesis wrote:
| Metric spaces are enriched categories. They are enriched over
| the positive reals. The 'hom' between a pair of points is
| then simply a number: their distance.
| drdeca wrote:
| And, these non-negative real numbers, which are these homs,
| are "hom objects", so regarded as objects in "the category
| with as objects the non-negative real numbers, and as
| morphisms, the 'being greater than or equal to' " ? Is that
| right?
|
| So, I guess, (\R_{>= 0}, >=, +, 0) is like, a monoidal
| category with + as the monoidal operation?
|
| So like, for x,y,z in the metric space, the
|
| well, from hom(x,y) and hom(y,z) I guess the idea is there
| is a designated composition morphism
|
| from hom(x,y) monoidalProduct hom(y,z) to hom(x,z)
|
| which is specifically,
|
| hom(x,y)+hom(y,z) >= hom(x,z)
|
| (I said designated, but there is only the one, which is
| just the fact above.)
|
| I.e. d(x,y)+d(y,z) >= d(x,z)
|
| (Note: I didn't manage to "just guess" this. I've seen it
| before, and was thinking it through as part of remembering
| how the idea worked. I am commenting this to both check my
| understanding in case I'm wrong, and to (assuming I'm
| remembering the idea correctly) provide an elaboration on
| what you said for anyone who might want more detail.)
| consilient wrote:
| > are "hom objects", so regarded as objects in "the
| category with as objects the non-negative real numbers,
| and as morphisms, the 'being greater than or equal to' "
| ?
|
| This works, but it's not quite what you want in most
| cases. There's a lot of stuff that requires you to enrich
| over a closed category, so instead we define `Hom(a,b)`
| to be `max(b - a, 0)` (which you can very roughly think
| of as replacing the mere proposition `a < b` with its
| "witnesses"). See https://www.emis.de/journals/TAC/reprin
| ts/articles/1/tr1.pdf for more.
| contravariant wrote:
| Indeed they are. I'm saying it may not be the right context
| in this case.
|
| At least what they seem to be doing has little to do with
| metrics, and a lot more to do with probability
| distributions.
| autopoiesis wrote:
| It's not clear what you're seeking. Probabilities appear
| because the magnitude of a space is a way of 'measuring'
| it -- and thus magnitude is closely related to entropy.
| Of course, you can follow your nose and find your way
| beyond mere spaces, and this may lead you to the notion
| of 'magnitude homology' [1]. But it's not clear that this
| generalization is the best way to introduce the idea of
| magnitude to ecology.
|
| [1] https://arxiv.org/abs/1711.00802
| [deleted]
| ssivark wrote:
| I found the paper easier to follow than the slides:
| https://www.maths.ed.ac.uk/~tl/mdiss.pdf
|
| (Less emphasis of the category theory, and more attention to the
| basic math behind entropy-like diversity measures)
|
| (TLDR) There are two important aspects:
|
| 1. Generalize diversity measures to when the categories are not
| fully distinct (as assumed for the Shannon entropy calculation)
| but have similarities parametrized in a "Z"-matrix here.
|
| 2. A parameter "q" to represent whether you value a category
| highly (for diversity purposes) even when it has only a single
| distinct example highly (q --> 0) or whether you value it highly
| only when it has many many examples (q --> infinity)
|
| * With Z = identity matrix (categories completely distinct) they
| show how different values of q reproduce different measures that
| have been considered before. (nicely summarized in a table)
|
| * _The generalization (parameterized by Z) when the categories
| can /do have some overlap is very elegant, and feels like an
| important step forward._ (especially how it makes the measure
| robust to how we partition a bunch of examples into distinct
| categories, so long as we keep track of the similarity between
| the categories)
|
| * The paper also summarizes a bunch of sensible properties that
| we would want any diversity measure to satisfy (like the one I
| just mentioned above).
|
| Fun stuff!
| circles_for-day wrote:
| This is the most confusing possible title. Why is it written this
| way?
| tgbugs wrote:
| Wow. The fact that there is an objective answer that is
| independent of any perspective on the importance of rare species
| is a rare gift, at least for this part of the problem.
|
| Some questions and thoughts.
|
| It seems that the result could vary based on how you construct
| the similarity matrix Z, e.g. is it purely taxonomic? or does it
| try to account for the ecological roles that a species is playing
| in the community, etc.
|
| A seeming limitation is that the optimization works only for a
| fixed set of n species. While it is useful for managing existing
| communities, it means that there is still a question of whether
| larger n is strictly better, and leaves open questions of how to
| deal with transient or migratory members (if the community is
| spatially bound).
|
| The answer I think, is that it depends on how the similarity
| matrix is constructed. If every species is fully dissimilar then
| increasing n is always a good thing. If you use niche space to
| construct it and new species do not some add or enter new niches
| so they overlap with others, then they will be close to another
| species in the matrix and increasing n will not have much impact.
| On the other hand if you use a purely taxonomic approach then you
| wind up balancing the number of birds and mammals regardless of
| niche.
|
| It is not clear to me whether it is possible to construct a
| similarity matrix that can account for the interaction between n,
| the carrying capacity of the ecosystem, and the number of
| available niches (or the ability of species to create new
| niches). By analogy if you have a stream (sunlight) powering
| water wheels, how many wheels and how many levels of gears
| (layers in the ecosystem) can be added, created, and/or
| sustained? At what point does adding an additional species mean
| that either two species are forced to be close together in the
| similarity matrix or both their populations must shrink in size
| because they must compete for the same energy sources?
|
| Does the model sometimes produce impractical results, e.g. that
| it is good to have a single member of a sexually reproducing
| species (this is probably an orthogonal concern and you would
| want to scale to real population sizes such that the minimum
| corresponded to the smallest viable a self sustaining
| population)?
|
| Is there evidence that maximizing diversity using this measure
| actually produces more robust and stable ecologies?
| tutfbhuf wrote:
| This presentation is getting into the idea of "magnitude" the
| invariant from category theory and how it can be used in
| mathematical ecology, especially when we're trying to max out
| diversity. It's put forward as a possible answer to a head-
| scratcher in mathematical ecology: How do we find the maximum
| diversity if we're given a list of n species and a similarity
| matrix?
|
| The presentation gives us a whole range of views on biodiversity,
| making it clear that we need to think about both common and rare
| species when we're figuring out diversity. It hints that this
| magnitude idea might give us a more detailed picture of diversity
| by considering how important different species are relative to
| each other. Then the presentation gets into the weeds of category
| theory, chatting about enriched categories and size-like
| invariants. It talks about stuff like monoidal categories,
| V-enriched categories, and linear categories. These ideas are
| brought up as tools to help us understand and calculate
| magnitude.
|
| The presentation also explores how these category theory ideas
| relate to metric spaces. It suggests that getting a handle on
| this relationship could give us even more insight into how to max
| out diversity. Lastly, the presentation brings up the Euler
| characteristic, which is a concept from algebraic topology. It
| hints that the Euler characteristic and magnitude are pretty
| tight, and understanding this connection could give us even more
| insight into how to max out diversity.
|
| So, to wrap it up, the presentation is a thorough look at how
| category theory ideas, especially magnitude, can be used to
| tackle problems in mathematical ecology. It suggests that these
| ideas can give us a more detailed understanding of diversity and
| might even help us figure out how to get the most diversity.
|
| References:
|
| https://arxiv.org/abs/2012.02113
|
| https://arxiv.org/abs/1606.00095
| seeknotfind wrote:
| Can someone tl;dr?
| [deleted]
| acmiyaguchi wrote:
| This is an interesting talk! I love reading about applications of
| math/computer science to ecology. The parent page has links to
| relevant papers that are worth reading, too. [0] The species
| similarity paper has some concrete examples on coral,
| butterflies, and gut microbiomes that I felt missing from the
| slides. [1]
|
| [0] https://www.maths.ed.ac.uk/~tl/genova/
|
| [1] https://www.maths.ed.ac.uk/~tl/mdiss.pdf
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