Plan (2021-12-07)
Theory: It looks like this proof is going to be a classic Horrible Analysis Proof. Fuck.
Applied: I got my gumbel softmax VAE running, now I have to run endless experiments
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@functional-analysis-done-okay
Plan (2021-12-07)
Theory: It looks like this proof is going to be a classic Horrible Analysis Proof. Fuck.
Applied: I got my gumbel softmax VAE running, now I have to run endless experiments

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Plan (2021-11-29)
Theory: I have one major element of the proof left: showing that if two probability distributions are a certain distance apart, they must have [desired property itās complicated]. I accidentally already showed the reverse direction (for distributions with bounded support, at least), because I confused myself.
Applied: Hmm. I still need to implement a gumbel softmax VAE, where by āimplementā I mean ācopy from github and get it to runā. I implemented a trivial environment and Iām not really seeing learning there, so clearly something is going wrong in the training alg.
Learning: Against my better judgment, Iām reading Kallenberg. Or rather Iām reading Kallenberg + corresponding chapters of Cinlar, because Iām not completely insane. Also a bunch of optimal transport stuff because Wasserstein distance.
Iām about 90% confident now that I will finish the entire proof. Iāve at least finished the first part, showing that conditional independence can be tested in a polynomial number of samples (if you make enough assumptions, conditional independence testing is impossible in generality)
I think Iām actually going to get the result I want. Finally. This is nice but it also makes me very unsure about continuing with grad school. It took me so long. So long that Iām embarrassed to say how long it took me.
Anyway I think this is evidence that Iām not cut out for this and I should consider dropping out.
The whole time Iāve been working on this proof, Iāve had it in the back of my mind that maybe I could just prove a bunch of general results and skip over all the details. And I kept saying āoh, no, thatās a mistake, Iāll end up wasting a ton of time looking for a general version and I should just prove it directlyā.
... and now Iāve done most of the direct proof, realized the general version, proved that and rewritten my proofs using the general thing.
Iām sure some of you are thinking āoh, well, you needed to do the direct versions in order to see the general versionā. Maybe. But Iād be a lot more confident of that if Iād tried and failed to find the general version first.

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Plan (2021-08-31)
Math: Iām closing in on the final result, very excited about it. I need to convert an existing result to my paradigm, prove an additional novel result, and then I think I should have everything modulo some cleanup
Code: Thereās one more code base I want to try adapting before itās back to the drawing board. I really hope it works but Iām worried it will not.
Plan (2021-08-16)
Theory: Thereās one or two major steps left in the proof. One of them should be easy, Iām just modifying an existing proof, but no one has written that proof down so instead Iām looking at a much more complicated proof and trying to adapt that.
Applied: Well, I am at least safe from being scooped because absolutely everyone else whoās looked at this problem has failed. Working with [redacted] is helpful, sheās much better at neural net engineering than I am
Learning: Still focusing on schemes and riemannian geometry
So there's lots of times when you need to learn everything actually useful on your own outside of official classes, eg cs you need to do some projects to learn to code and classes are insufficient.
Is US math grad school like that? Specifically the divide between knowing baby algebra/topology/ a little homology/homotopy and being like capable of doing algebraic topology research? Or the like for other areas of math.
Because the gap seems very wide from here and it would suck if I discovered in two years that I was supposed to have bridged it largely on my own.
My experience (which is in the cs department, not the math one, but I expect it to be similar) is yes. There are advanced prob/stats/analysis courses available, but the particulars for my research are not covered and it was really just down to me to read papers and relevant textbooks.
Is the expectation that your advisor points you to the textbooks and papers, or that you independently find the papers/texts and the time for it?
Neither/both. Ideally your supervisor will point you towards interesting things, but itās expected that you will be going out and actively looking yourself as well.
My research project is outside either of my supervisorsā fields of expertise, so I think I probably do more self-directed work than is usual, but I would expect to do at least some no matter what, especially for learning the prerequisites.
So there's lots of times when you need to learn everything actually useful on your own outside of official classes, eg cs you need to do some projects to learn to code and classes are insufficient.
Is US math grad school like that? Specifically the divide between knowing baby algebra/topology/ a little homology/homotopy and being like capable of doing algebraic topology research? Or the like for other areas of math.
Because the gap seems very wide from here and it would suck if I discovered in two years that I was supposed to have bridged it largely on my own.
My experience (which is in the cs department, not the math one, but I expect it to be similar) is yes. There are advanced prob/stats/analysis courses available, but the particulars for my research are not covered and it was really just down to me to read papers and relevant textbooks.
Plan (2021-07-19)
Theory: Still havenāt figured out the blocked part of the proof. Specifically thereās a claim that a particular estimator produces a hypothesis test for conditional independence with particular power, but I honestly canāt see it (itās certainly in the realm, youāre estimating a thing that is 0 if conditional independence holds, but itās a continuous variable so I donāt see how we can get non-zero measure on it actually being zero)
Code: Doing some devops, mostly have things set up nicely. Still need to clean the actual code up and refactor it to fit into my overall framework.
Learning: Iām going to start recording some of the things Iām focusing on learning. Mostly this is stuff that sits in between direct research relevance (which would go under theory) and just for fun (which I might post about anyway)
Anyway this week Iām going to read through some literature on neural architecture search, and finish going through Ideals, Varieties, and Algorithms.
Code: several of the papers I was looking at turned out to be useless (solving the wrong problem), so getting them running was a big waste of time
I got through a bunch of NAS and most of ideals varieties and algorithms

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Plan (2021-07-19)
Theory: Still havenāt figured out the blocked part of the proof. Specifically thereās a claim that a particular estimator produces a hypothesis test for conditional independence with particular power, but I honestly canāt see it (itās certainly in the realm, youāre estimating a thing that is 0 if conditional independence holds, but itās a continuous variable so I donāt see how we can get non-zero measure on it actually being zero)
Code: Doing some devops, mostly have things set up nicely. Still need to clean the actual code up and refactor it to fit into my overall framework.
Learning: Iām going to start recording some of the things Iām focusing on learning. Mostly this is stuff that sits in between direct research relevance (which would go under theory) and just for fun (which I might post about anyway)
Anyway this week Iām going to read through some literature on neural architecture search, and finish going through Ideals, Varieties, and Algorithms.
Plan 2021-07-06:
Overall Iām v happy with progress on theory and code
Theory: Have most of the main proof done. Need to send an email to [the scary prof] with a question about his field. Need to write up the last bit of the proof for the noise terms. Need to formalize everything.
Code: I picked up a collaborator (my girlfriend who wants to publish RL papers). Sheās way more motivated about writing ML code than I am, which is extremely helpful because motivation is my biggest block by far. Also helpful because it forces me to be more organized since now I have another coder to direct. Weāre working through some reimplementations of relevant work. Also she has a 3090* which is extremely useful for experiments.
* for the pure math people a 3090 is more or less the most powerful consumer GPU you can get, and thereās a major shortage of GPUs right now so I havenāt upgraded my own.
I finished the proof and sent the scary email! I still need to show that a particular term is/is bounded by a polynomial, but Iām reasonably confident thatās true and should be easy.
Iām somewhat blocked on the response from the scary prof now, so Iām just going to be doing code for a while.
Plan 2021-07-06:
Overall Iām v happy with progress on theory and code
Theory: Have most of the main proof done. Need to send an email to [the scary prof] with a question about his field. Need to write up the last bit of the proof for the noise terms. Need to formalize everything.
Code: I picked up a collaborator (my girlfriend who wants to publish RL papers). Sheās way more motivated about writing ML code than I am, which is extremely helpful because motivation is my biggest block by far. Also helpful because it forces me to be more organized since now I have another coder to direct. Weāre working through some reimplementations of relevant work. Also she has a 3090* which is extremely useful for experiments.
* for the pure math people a 3090 is more or less the most powerful consumer GPU you can get, and thereās a major shortage of GPUs right now so I havenāt upgraded my own.
Speaking of algebra, this is maybe quite nice for pedagogy: https://nathancarter.github.io/group-explorer/index.html
(Itās a website explaining many of the common finite groups, defining and visualizing them in various ways)
i miss algebra

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Plan (2021-05-18):
Theory: Need to prove bounds on the error terms
Applied: still feeling very averse to writing code, but it got a little better this week.
Slutskyās Theorem
This is a nice theorem I learned recently and Iām surprised I never learned it before.
If $X_n$ is a sequence of random variables converging in distribution to $X$, and $Y_n$ is a sequence of random variables converging in probability to a constant $c$, then:
$X_n + Y_n$ converges in distribution to $X + c$
$X_n Y_n$ converges in distribution $cX$
$X_n / Y_n$ converges in distribution to $X / c$
Yeah.
Like, you know the sample mean has a normal distribution for large samples, by the central limit theorem.
but what about the t statistic: sample mean over estimated standard error?
By the third condition above, that also appraoches a normal distribution. Thus the t test and z test will give the same answers for large samples
z statistic:
M / (Ļ/ān)
where M is the sample mean. Approaches standard normal distribution, by central limit theorem.
t statistic:
M / (S/ān)
where S is the sample standard deviation. S approaches Ļ, so by Slutskyās theorem, this approaches the same distribution as the z statistic.