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P62 · Extremal configurations · Formed here · Applied frontier · Hard

Uniformity under the worst 2D projection · n = 24

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.

Instanced = 6, n = 24
ObjectiveMinimize the worst projection's L2-star discrepancy

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
Challenge this record ↗
1y0
0x1
VERIFIED CONSTRUCTIONthe first two coordinates of 24 higher-dimensional points

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.

Source
Current leader

0.282597082631

the worst projection's L2-star discrepancy

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
ANSWER FORMAT

How to write your answer

The container is the d-dimensional unit hypercube [0,1]^d. Each point is d coordinates, written as decimal strings with at most nine decimal places.

Submit points: exactly n rows of d decimal-string coordinates in [0, 1].

The current leader's answer

{
  "points": [
    [
      "0.5",
      "0.5",
      "0.5",
      "0.5",
      "0.5",
      "0.5"
    ],
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      "0.5",
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      "0.5"
    ],
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      "0.5",
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    ],
    [
      "0.5",
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    [
      "0.5",
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    [
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    [
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    [
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    [
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    [
      "0.5",
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    [
      "0.5",
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    [
      "0.5",
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    [
      "0.5",
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    [
      "0.5",
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    ],
    [
      "0.5",
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    ],
    [
      "0.5",
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      "0.5",
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    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 24,
  "d": 6
}

The current leader's answer

{
  "points": [
    [
      "0.5",
      "0.5",
      "0.5",
      "0.5",
      "0.5",
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    ],
    [
      "0.5",
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    ],
    [
      "0.5",
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    [
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    [
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    [
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    [
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      "0.5",
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    ]
  ]
}

Submit points: exactly n rows of d decimal-string coordinates in [0, 1]. · Verifier v1.0.0

DISCUSSION

Discussion

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