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> I parameterized the wheel as a twenty sided polygon with a variable radius for each vertex, ri. Could this way of representing the problem (radius) introduce

> I parameterized the wheel as a twenty sided polygon with a variable radius for each vertex, ri. Could this way of representing the problem (radius) introduce bias to the solution? It could probably make many kind of shapes, but it may need to overcome some huge local maxima in order to drastically change it to something else by tweaking multiple of the radii. When writing my thesis on spoke patterns for bicycle wheels, one of the representations outperformed the others, as it had symmetry baked in. That could have made it not find some esoteric solutions (which is why I had multiple representations).

anticristi 6y ago
I totally agree! It would have been better to avoid bias by using a pixel/voxel representation (1=material present, 0=no material present), like Airbus did with wing structure optimization: https://backend.orbit.dtu.dk/ws/portalfiles/portal/138121106/ppnature23911_Aage_ORBIT.pdf https://backend.orbit.dtu.dk/ws/portalfiles/portal/138121106... (Not claiming that OP's work isn't absolutely cool, rather suggesting paths for future improvements. :D)
- anticristi 6y ago
  @matsemann Would you mind sharing your work?
  - matsemann 6y ago
    It was discussed here[0] at the time :) [0]: https://news.ycombinator.com/item?id=10410813 https://news.ycombinator.com/item?id=10410813
- matsemann 6y ago
  On/off representations are cool, but also a bit harder to work with. One can end up having disjoint groups that don't really make sense, so lots of permutations has to be thrown out. So finding the correct representation can actually be very hard.
  - neutronicus 6y ago
    The problem you're describing is kind of dual to the problem with parametrized representations. It's easy to construct parametrized boundary representations (as OP used) that maintain topological invariants, but it's very difficult to use them to explore the space of possible topologies (which would be relevant if OP wanted to say, design a lighter wheel).
bluesign 6y ago
I think actually ri doesn’t introduce bias, maybe N=20 sides can be problem for bias, sufficiently large N should not cause problem.
- Retric 6y ago
  The issue is the points are all equally spaced. Working back from a circle that’s not an issue, but suppose the optimal solution had varying side lengths. You could approximate that with very large n, but that’s about it.
raverbashing 6y ago
> Could this way of representing the problem (radius) introduce bias to the solution? Yes, but you're always going to have a prior. You could model it as, let's say, a shape in a square grid with a certain area. Or a collection of line segments that are closed. This is an issue common to most optimization problems, you want the simplest model that will be efficient, otherwise you might risk overfitting (or just taking too long to optimize)
ssivark 6y ago
Responding broadly to this and other sibling comments about different representations for optimization... This can be considered a generalized example of the bias-variance tradeoff. Imposing symmetry or a parametric form can be considered forms of regularization/bias that help converge to a well-formed solutions.

anticristi 6y ago

I totally agree! It would have been better to avoid bias by using a pixel/voxel representation (1=material present, 0=no material present), like Airbus did with wing structure optimization: https://backend.orbit.dtu.dk/ws/portalfiles/portal/138121106/ppnature23911_Aage_ORBIT.pdf https://backend.orbit.dtu.dk/ws/portalfiles/portal/138121106... (Not claiming that OP's work isn't absolutely cool, rather suggesting paths for future improvements. :D)

anticristi 6y ago
@matsemann Would you mind sharing your work?
- matsemann 6y ago
  It was discussed here[0] at the time :) [0]: https://news.ycombinator.com/item?id=10410813 https://news.ycombinator.com/item?id=10410813
matsemann 6y ago
On/off representations are cool, but also a bit harder to work with. One can end up having disjoint groups that don't really make sense, so lots of permutations has to be thrown out. So finding the correct representation can actually be very hard.
- neutronicus 6y ago
  The problem you're describing is kind of dual to the problem with parametrized representations. It's easy to construct parametrized boundary representations (as OP used) that maintain topological invariants, but it's very difficult to use them to explore the space of possible topologies (which would be relevant if OP wanted to say, design a lighter wheel).

bluesign 6y ago

I think actually ri doesn’t introduce bias, maybe N=20 sides can be problem for bias, sufficiently large N should not cause problem.

Retric 6y ago
The issue is the points are all equally spaced. Working back from a circle that’s not an issue, but suppose the optimal solution had varying side lengths. You could approximate that with very large n, but that’s about it.

raverbashing 6y ago

> Could this way of representing the problem (radius) introduce bias to the solution? Yes, but you're always going to have a prior. You could model it as, let's say, a shape in a square grid with a certain area. Or a collection of line segments that are closed. This is an issue common to most optimization problems, you want the simplest model that will be efficient, otherwise you might risk overfitting (or just taking too long to optimize)

ssivark 6y ago

Responding broadly to this and other sibling comments about different representations for optimization... This can be considered a generalized example of the bias-variance tradeoff. Imposing symmetry or a parametric form can be considered forms of regularization/bias that help converge to a well-formed solutions.