131–140 of 167 posts
Once you start looking at the world through the lens of frequency domain a lot of neat tricks become simple. I have some demo code that uses fourier transform on webcam video to read a heartrate off a person's face, basically looking for what frequency holds peak energy.
I don't think pulsing skin (due to blood flow) is visible from a webcam though.
People go all dopey eyed about "frequency space", that's a red herring. The take away should be that a problem centric coordinate system is enormously helpful. After all, what Copernicus showed is that the mind bogglingly complicated motion of planets become a whole lot simpler if you change the coordinate system. Ptolemaic model of epicycles were an adhoc form of Fourier analysis - decomposing periodic motions over…
It’s kind of intriguing that predicting the future state of any quantum system becomes almost trivial—assuming you can diagonalize the Hamiltonian. But good luck with that in general. (In other words, a “simple” reference frame always exists via unitary conjugation, but finding it is very difficult.)
It's disconcerting at times, the scope of finite and infinite dimensional linear algebra, especially when done on a convenient basis.
My favorite story about the Fourier Transform is that Carl Friedrich Gauss stumbled upon the algorithm for the Fast Fourier Algorthim over a century before Cooley and Tukey’s publication in 1965 (which itself revolutionized digital signal processing).[1] He was apparently studying the motion of the asteroids Pallas and Juno and wrote the algorithm down in his notes but it never made it into public knowledge. [1] http…
How was Gauss so productive with 6 children?
People go all dopey eyed about "frequency space", that's a red herring. The take away should be that a problem centric coordinate system is enormously helpful. After all, what Copernicus showed is that the mind bogglingly complicated motion of planets become a whole lot simpler if you change the coordinate system. Ptolemaic model of epicycles were an adhoc form of Fourier analysis - decomposing periodic motions over…
All these transforms are switching to an eigenbasis of some differential operator (that usually corresponds to a differential equation of interest). Spherical harmonics, Bessel and Henkel functions, which are the radial versions of sines/cosines and complex exponential, respectively, and on and on. The next big jumps were to collections of functions not parameterized by subsets of R^n. Wavelets use a tree shapes para…
For GPT like models, I see sentences as trajectories in the embedded space. These trajectories look quite complicated and no obvious from their geometrical stand point. My hope is that if we get the coordinate system right, we may see something more intelligible going on.
This is just a hope, a mental bias. I do not have any solid argument for why it should be as I describe.
Earlier quoted context omitted.
All these transforms are switching to an eigenbasis of some differential operator (that usually corresponds to a differential equation of interest). Spherical harmonics, Bessel and Henkel functions, which are the radial versions of sines/cosines and complex exponential, respectively, and on and on. The next big jumps were to collections of functions not parameterized by subsets of R^n. Wavelets use a tree shapes para…
Note that I'm not great at math so it's possible I've entirely misunderstood you. Here's an example of directly leveraging a transform to optimize the training process. ( https://arxiv.org/abs/2410.21265 ) And here are two examples that apply geometry to neural nets more generally. ( https://arxiv.org/abs/2506.13018 ) ( https://arxiv.org/abs/2309.16512 )
Earlier quoted context omitted.
> I was minimizing time so spent while she was minimizing number of washes I'm probably just slow, but I'm not following. Do you mean because you went fast, you had to run another cycle to clean everything properly? If you haven't already, you should watch the Technology Connections series on dishwashers. https://www.youtube.com/watch?v=jHP942Livy0
Since I’m rushing to load it as fast as possible the packing is not as good as hers so some dishes are left out. Overall this leads to more loads.
Okay, who's gonna write the story > The unreasonable effectiveness of The Unreasonable Effectiveness title?
Earlier quoted context omitted.
I posit Gauss knew he was a GOAT and had ego. But I also posit he loved his children. So, a nepotistic delivery was beneficial for his family, and advising his son to seek excellence outside the shadow cast by Gauss himself wasn't stamping on dreams (in my view) it was seeking the happiest outcome. Without overdoing it, the suicide rate for rich kids with famous parents isn't nothing. There are positive examples, Ste…
What does "She isn't wings" mean?
Too bad -- the article doesn't mention Gauss. The Fourier transform is best presented to students in its original mathematical form, then coded in the FFT form. It also serves as a practical introduction to complex numbers. As to the listed patent, it moves uncomfortably close to being a patent on mathematics, which isn't permitted. But I wouldn't be surprised to see many outstanding patents that have this hidden pro…
Pretty sure in the USA you can patent mathematics if it is an integral part of the realisation of a physical system.* There is a book "Math you Can't Use" that discusses this. * not a legal definition, IANAL.
Yes, that's true. In that example, you're not patenting mathematics, you're patenting a specific application, which can be patented. In my reading I see that mathematics per se is an abstract intellectual concept, thus not patentable (reference: https://ghbintellect.com/can-you-patent-a-formula/).
There is plenty of case law in modern times where the distinction between an abstract mathematical idea, and an application of that idea, were the issues under discussion.
An obligatory XKCD reference: https://xkcd.com/435/
And IANAL also.