Isochronous Curves
en.wikipedia.org
en.wikipedia.org
T = 2*pi*sqrt(L/g) = 2*pi*sqrt(1 m / 9.81 m/s^2) = 2.01 s>To make its period isochronous, Huygens mounted cycloidal-shaped metal 'chops' next to the pivots in his clocks, that constrained the suspension cord and forced the pendulum to follow a cycloid arc.[53] This solution didn't prove as practical as simply limiting the pendulum's swing to small angles of a few degrees.
Seriously though, I would be weirdly fascinated to see what else you guys subscribe to and if there are any correlations to my subscriptions. I find this channelling people into little boxes fascinating, and those curves are also very cool.
It would be interesting to me to know how many stories in HN trend simply because a group of people were all recommended the same content.
It is kind of uncanny that we all have seen this particular video in our recommendations recently. It makes me want to try and create a graph of the connections between all the links.
Additionally, if youtube was forced to only play videos where copyright was proven to belong to the owner of the channel, would it collapse completely.
Is that perhaps what is shrinking the bag of marbles ?
Which interestingly suggests that if we force youtube to actually respect IP, it would reward new creators of quality work more proportionately
Freedom is Slavery
Ignorance of our algorithms is necessary
But I am a little surprised by the amount of people that get that from the recommender system too.
Feynman's Lost Lecture (ft. 3Blue1Brown) [video] https://news.ycombinator.com/item?id=17927217
The framework is basically minimization in functional space (as opposed to R^n or a subset thereof, the more common case).
The (to me) surprising thing is that finding an extremum in functional space (i.e. infinite-dimensional space) can be reduced to solving a differential equation, via Euler–Lagrange.
The first problem feels intractable, while the second is "just" numerical integration.
They clearly aren't, especially when it comes to proving some sort of formal property about them (which is something a mathematician would worry about, but which is rarely an engineer's first concern: they'd only worry about that type of thing when numerical integration starts producing "crazy" results).
However, there is a very intuitive way to compute an approximate solution to a system of PDE's, namely increase time in very tiny steps and solve the resulting system of equations each time, rinse and repeat.
Now, when you deal with arbitrary functions, there are so many ways to "represent" them: as various kind of series, as algebraic composition of elementary functions, as solutions to implicit equations, as solutions to PDE's, as polynomial approximations, as neural networks, etc ...
These representations are usually dense in "regular" functional spaces and you do get a lot of mileage out of them.
Whichever way you chose to represent functions in your functional space, the problem of finding an extrema over that representation is very much not obvious in the calculus of variation setting (or even that the extremum will be itself be representable, for that matter), whereas the conversion to a system of PDE gives you a clear path to victory - as long as the representation can be differentiated.
A good way to think about problems such as these are to look at the extreme cases. In this case, consider a flat plane: the ball takes infinite time, since it never rolls at all. And the fastest path would be a valley (though, as far as being v-shaped as possible), with the ball accelerating part of the way and decelerating at the end, spending the majority of the time at a high speed.
I’ve worked as a practitioner in computer vision as well and I think I disagree with your “under taught” comment.
In practice, the variational methods are not competitive at all with deep learning models, and even very naive transfer learning can apply better inpainting or segmentation than the older variational methods.
It is already merely a footnote in the history of computer vision. In 10 years it won’t even merit being part of literature reviews.
So for many applied domains of ML, I think spending roughly zero time on these variational methods in schooling is a good idea. You can always come back to them later, but you probably won’t need to.
A different application domain, variational Bayesian inference, might be a case where there is more reason to care. Although, things like ADVI are more about one very fixed approach to the problem (minimizing KL divergence to find a best approximating distribution from a class with known differentiability properties), and only in novel research would knowledge of general variational techniques give much value.
I really don't see why these two things need be pitted against each others.
What you're describing (Euler-lagrange applied to computer vision problems) is a direct application of the calculus of variation to the image itself.
It's rather naive because it uses an extremely weak representation of the content of the image (i.e. a bunch of pixels which essentially means working with a piecewise-constant basis of the functional space), and as you noted, it is easily beaten by modern techniques.
However: Deep neural nets are nothing but another, much more powerful "basis" of functional space (I put "basis" in quote because they're not linear. OTOH, they're generative).
A deep net is a finite set of real numbers that maps to a point in functional space, and any arbitrary continuous maps from R^n to R^p can be approximated by deep nets given enough weights and neurons.
So, if deep nets are dense in functional space, and "training" a deep net is nothing but solving a minimization problem in functional space, then I don't see why the "calculus of variation" ==> "PDE" via Euler-Lagrange route can't bee applied to training deep nets instead of backprop-based minimizers.
Now, if you're claiming that backprop works better than solving the PDE induced by Euler-Lagrange on the functional minimization problem of training a deep-net, I'm very happy to believe you, but I've yet to see literature demonstrating this.
It’s essentially a way to encode semantics as prior distributions over possible images, then variational methods “optimize” for the posterior parameters given some data.
I’m trying to say that that whole exercise listed above, conceptually, is basically a failed idea compared with solving the same problems using deep learning to automatically discover what that manifold looks like in parameter space and transform it directly to the output space of practical interest (segmentation boundaries, etc.).
Fair enough, but my main point stands: has anyone tried to apply Euler-Lagrange and a PDE solver to the problem of training a deep net or is it obvious that backprop-based methods will work better?
“When I was in high school, my physics teacher—whose name was Mr. Bader—called me down one day after physics class and said, ‘You look bored; I want to tell you something interesting.’ Then he told me something which I found absolutely fascinating, and have, since then, always found fascinating. Every time the subject comes up, I work on it. In fact, when I began to prepare this lecture I found myself making more analyses on the thing. Instead of worrying about the lecture, I got involved in a new problem. The subject is this—the principle of least action."
The rest at: http://www.feynmanlectures.caltech.edu/II_19.html
- principle of least action (lagrangians)
- adiabatic cycles in thermodynamics
- quantum mechanics
- etc..
Looks like it's a fairly fundamental thing in the world.
The only way to find out is to click the link.
edit: And it's legit!
Which to be fair, I'm entirely fine with as long as the projects are legitimate. In the world of programming, there are so many cool and useful libraries, services and tools out there that you have little hope to find out about them otherwise.
( Shameless plug - an open source implementation of it: https://github.com/graphhopper/graphhopper#for-analysis )