Uniformity under the worst 2D projection
Place n points in [0,1]^d. For every pair of coordinates (r, s), keep just those two coordinates to get a planar projection and score it with P59's exact L2-star formula; the score is the largest over all C(d,2) projections. Make it as small as possible.
Formal definition
- ContainerThe d-dimensional unit hypercube [0,1]^d
- SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
- ObjectiveMinimize the maximum L2-star discrepancy over all two-coordinate projections; the C(d,2) projections share one denominator, so the maximum is taken exactly on integer numerators
- ScoringThe record is the worst projection's exact integer with the denominator cleared; the page shows that projection's discrepancy, rounded up at the twelfth decimal
Getting a feel for it
Why stare at projections
A fine full-dimensional score cannot stop two particular columns from striping when seen together. Computer experiments, rendering and QMC are dominated by low-order interactions, and the worst 2D projection is the first mirror to crack.
Where the frontier is
Projection uniformity is an active direction in experimental design, but the continuous worst-projection objective has no published table of optima for fixed (n, d). Every sub-problem is open.
SourceCurrent best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
Discussion (0) ↓Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
No threads yet. Be the first to talk about this problem?