extremelearning

9 ms·

1. 31.▲
   by extremelearning 6y ago

   Thanks! To be honest, I hadn't previously made it prominent as I thought RSS had basically died several years ago. However, based on the multiple comments in this parent thread, evidently RSS is still alive and well. So I'll defi
2. 32.▲
   by extremelearning 6y ago

   I am a statistician / data scientist who blogs about nifty sampling methods, which are frequently used in rendering computer graphics such as: ray-tracing (via Quasi-Monte Carlo methods), object placement, dithering, etc... http:&#x2F
3. 33.▲
   by extremelearning 8y ago

   i suspect a caching issue. I cleared my wordpress cache, so hopefully it will appear correct to others soon. ;)
4. 34.▲
   by extremelearning 8y ago

   Yes. Consider π. For any S>0, you can construct an infinite number of rational approximations that have a score of less than S. But for any quadratic irrational (surd), as the depth of the corresponding continued fraction increases, the
5. 35.▲
   by extremelearning 8y ago

   Good question, but I don't know and haven't really done anything substantially related that might even give us a hint. Hopefully someone else chime in on this thread. ;)
6. 36.▲
   by extremelearning 8y ago

   I pondered this issue for what seemed like an inordinate amount of time namely: on how to describe this subtle but key difference. Unfortunately, I couldn't find a nice way, so i glossed over this point, which you correctly say makes m
7. 37.▲
   by extremelearning 8y ago

   @svat's comment and link may also be helpful in this regard.
8. 38.▲
   by extremelearning 8y ago

   Thanks. No, you aren't crazy, but maybe my typos made you crazy! ;) I have now fixed a couple of typos in the grammar and continued fractions expressions for that section.
9. 39.▲
   by extremelearning 8y ago

   Generally my answer is that this is for the same reason that fitting lines of best fit to data is nearly always done via a least-squares fitting. Squaring has a few major benefits. The first is that is never negative. Therefore, one might a
10. 40.▲
    by extremelearning 8y ago

    The distance from Angkor Wat to Giza is: 4754 miles. And the distance from Gize to Nazca is: 4754 x φ miles. By adding these, you get that the distance from Angkor Wat to Nazca is 4754 (1+φ) miles. But φ is defined such that 1+φ = φ², [Ver
11. 41.▲
    by extremelearning 8y ago

    Absolutely! Everybody loves the numberphile videos. They frequently distil deep maths topics into very intuitive and visual explanations. ;)
12. 42.▲
    by extremelearning 8y ago

    What I find fascinating is that there seem to be so many valid ways to generalize the Golden Ratio. As you say, the "metallic means" [1] are quite well-known, and relate to the recurrence relation via: T(n) = m *T(n-1)+ T(n-2), fo
13. 43.▲
    by extremelearning 8y ago

    LOL! You're totally right. It should be 85/10 and 425/50. Now fixed.
14. 44.▲
    by extremelearning 8y ago

    Author here. Happy to try to answer any questions any one might have on this post or topic. )
15. 45.▲
    by extremelearning 8y ago

    Although using a low discrepancy sequence (eg Halton, Sobol, R2) might be a good starting point, these sequences are designed to maximize the distance between points assuming that the left-and-right edges wrap, as well as the top-and-bottom
16. 46.▲
    by extremelearning 8y ago

    1. Stretching I think you and Jacob are on to something when we realise that the R_2 sequence (and presumably R_D) loses some of its properties when it is naively stretched. I think if we solve this problem then we can make some real progre
17. 47.▲
    by extremelearning 8y ago

    Sobol's sequence is really quite amazing. However, there are three reasons why it doesn't get as much attention as it probably deserves. First is that the maths is far more complex than that of other sequences. Here is a the easie
18. 48.▲
    by extremelearning 8y ago

    That's great. I'm glad you got it working. ;)
19. 49.▲
    by extremelearning 8y ago

    Yes. I think that using the dither matrix based on the R_2 sequence should be competitive with any of the the dither masks mentioned on that link. Thus in situations like video, and/or ray tracing where computing speed is of the essenc
20. 50.▲
    by extremelearning 8y ago

    Thank you for your kind words! Regarding the mapping of R_2 to the surface of the sphere, I totally agree that although it is better than other options, it is far from optimal. Also, yes, these visualizations are looking from the North pole
21. 51.▲
    by extremelearning 8y ago

    That's strange. To double-check that i typed it into my blog correctly, I copy-and-pasted the code from the post into my python IDE and it compiles and runs properly in my environment. For what it is worth, I am using Python 2.7 on a
22. 52.▲
    by extremelearning 8y ago

    I have now inserted some sound demonstrations into my blog post! You can now listen to what 1-dimensional (mon) and 2-dimensional (stereo) quasirandom sound might be like. I will leave it to you to decide which ones you prefer, and maybe ev
23. 53.▲
    by extremelearning 8y ago

    Unfortunately, I haven’t found a nice solution to this problem yet. For those requiring quasirandom Monte Carlo integration over a say a 2x1 rectangular grid there is always the option of scaling the unit square points by 2 and then reject
24. 54.▲
    by extremelearning 8y ago

    Thanks for this advice. I am new to blogging and certainly new to HN front page etiquette. I think my excitement got the better of me! I have now taken it off.
25. 55.▲
    by extremelearning 8y ago

    Only blue noise... I have absolutely no idea what it would sound like. So that means I’ll definitely try it later tonight and update the blog post. Tx
26. 56.▲
    by extremelearning 8y ago

    Thanks for the kind words and promotion! ;) These findings were a combination of a lot of literature searching, perseverance, decades of mathematical intuition and computer modelling of various hypotheses. I had a problem to solve so I was
27. 57.▲
    by extremelearning 8y ago

    For a known fixed value of n, you can get amazing convergence rates. For example, the basic midpoint rule gives O(1/n^2) and Simpson's rule gives O(1/n^4). Despite this, there are two major use cases for quasirandom sequences
28. 58.▲
    by extremelearning 8y ago

    Yes, using an open (infinite) one dimensional low-discrepancy sequence in conjunction with a space-filling curve such as the Hilbert-curve has been explored. For example by Owen [1] Niederreiter himself has a beautiful paper that includes a
29. 59.▲
    by extremelearning 8y ago

    My intent when I wrote that phrase was two-fold. Firstly to emphasise the large body of work that was done by Weyl, Kronecker, Hurwitz et al relating to the equidistribtuion theorem, badly approximately numbers and diophantine approximatio
30. 60.▲
    by extremelearning 8y ago

    Thanks tlb. You're totally right. I wrote the example code to match the notation used in the post which is a reflection of my maths background. I suspect that from a programming perspective using the recurrence relation: z[i+1] = (z[i]