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Aidan Rocke
@
bayesianbrain
Europe
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An applied mathematician working on morphogenesis and foundations for intelligent behavior. // Rules for scientific discourse: bit.ly/2S4ezni
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Osobe koje vas prate
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Aidan Rocke
@bayesianbrain
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16 h |
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'Fractional Calculus and Variational Mechanics' //large.stanford.edu/courses/2007/p…
An explanation of how the fractional calculus allows an extension of the Lagrangian to non-conservative systems.
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Trends in Cognitive Sciences
@TrendsCognSci
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26. stu |
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Bridging Motor and Cognitive Control: It’s About Time!
@harrison_ritz, @amitaishenhav, & Romy Frömer highlight recent @NatureNeuro work revealing similarities in the algorithms that control our thoughts and movements
@mjaztwit, @EggerSeth cell.com/trends/cogniti…
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Aidan Rocke
@bayesianbrain
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18 h |
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Polymath is not quite Manhattan-style. :) For the Manhattan project you have hundreds of incredibly stressed scientists working on an existential crisis, not a nice theoretical problem.
The polymath effort was also started by scientists with zero relevant prior experience.
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Aidan Rocke
@bayesianbrain
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19 h |
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I also don't know if I am too nice but I think you are much too modest. I think a lot of your ideas are pretty cool. :)
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Aidan Rocke
@bayesianbrain
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19 h |
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The thing is nobody has the answer to this question so diversity of thought is more important than a consensus established by a select few. As for elitism, I believe we should take it from nobody.
I might add that we are all here to learn from each other right? :)
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Aidan Rocke
@bayesianbrain
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1. velj |
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@ampanmdagaba, @NoahGuzman14 do you guys think this can work for problems in theoretical neuroscience?
I mean interesting problems which won't be worked out in the next 10 years otherwise as they simultaneously require a combination of skills and solving a coordination problem.
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Aidan Rocke
@bayesianbrain
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1. velj |
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'The Polymath Project is a collaboration among mathematicians to solve important and difficult mathematical problems by coordinating many mathematicians to communicate with each other on finding the best route to the solution.'
link: polymathprojects.org
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Aidan Rocke
@bayesianbrain
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I think that for very abstract problems like this it helps to work on concrete problems to develop an intuition. That would be my approach at least.
Then you could find the common thread between different problems, a unifying perspective.
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Aidan Rocke
@bayesianbrain
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Aha, this is an interesting idea. :)
This is probably the reason why deep learning works on many different data sets. You might want to checkout this paper: arxiv.org/abs/1608.08225
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Aidan Rocke
@bayesianbrain
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This isn’t a very formal argument but even if I formalise it, I doubt this is half the picture.
Interesting research has been done on the probability of chaos in higher-dimensional dynamical systems for example: arxiv.org/pdf/nlin/04080… which is probably very relevant. 3/3
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Aidan Rocke
@bayesianbrain
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1. velj |
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So even if it may be embedded in a more complex and higher-dimensional dynamical system, its proper phase-space probably has much lower dimension relatively speaking. 2/3
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Aidan Rocke
@bayesianbrain
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1. velj |
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Here's my take on the question so far:
If a system is stable it must obey a homeostatic principle and therefore be controllable. Controllability also entails that it has a causal description in terms of local interactions. 1/3
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Aidan Rocke
@bayesianbrain
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1. velj |
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You could define an equivalence relation on all dynamical systems where you map a dynamical system to the dimension of its phase space. If you order these systems in terms of their dimension you will find that you don't have an upper-bound.
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Aidan Rocke
@bayesianbrain
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Do you mean an intrinsic dimension for particular data or all data? If the former then that's the basis of manifold learning, finding the intrinsic dimension of the data.
If you mean an intrinsic dimension for all data I am not sure that would work.
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Aidan Rocke
@bayesianbrain
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1. velj |
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Physical processes which may be described using a system of ODEs or PDEs.
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Aidan Rocke
@bayesianbrain
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1. velj |
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Is there a part of my question which is unclear? Or is there is a false axiom embedded in my question?
I actually wonder why I haven't seen any papers on the Manifold Hypothesis which directly attempt to address this question.
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Aidan Rocke
@bayesianbrain
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1. velj |
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Might there be a reason why stable physical processes would tend to have low-dimensional phase spaces? 2/2
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Aidan Rocke
@bayesianbrain
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1. velj |
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Here's the problem in two parts:
In order to do ML we need to collect a lot of data from a data-generating process so this process must be stable. Empirically we observe that the intrinsic dimension d of the data is generally much smaller than the ambient dimension D.
1/2
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Aidan Rocke
@bayesianbrain
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1. velj |
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Note: I think this might be one of the most interesting open problems in machine learning and neural information processing, unless a theoretical neuroscientist has already adequately addressed this question in a slightly different setting.
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Aidan Rocke
@bayesianbrain
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Finally, we are all on Twitter to exchange ideas and not one-up each other so I hope everyone feels free to share their perspective. :)
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