P65 · Extremal configurations · Formed here · Applied frontier · Hard

The most erasure-robust measurement directions · n = 12

Choose n nonzero vectors in R^d. For every subset of size d, take the squared determinant of the matrix with those vectors as columns, divided by the product of their squared norms; maximize the minimum of this normalized volume over all subsets. Zero means some d surviving measurements cannot recover the space at all.

Instanced = 4, n = 12

ObjectiveMaximize the worst subset's normalized volume

Formal definition

ContainerReal d-space R^d; an answer is n measurement directions
SubmissionExactly n nonzero vectors with d coordinates each
ObjectiveMaximize min det(V_S)²/Π|v_i|² over all C(n,d) subsets; determinants and norms are rational, compared by cross-multiplication
WordingThe normalized volume is a robustness proxy aligned with numerical stability, not the optimal reconstruction error under every noise model; this problem claims maximin volume and nothing more
ScoringThe record is floor(min normalized volume · 10¹⁸), rounded against the submitter; the page shows that volume, rounded down at the twelfth decimal

Challenge this record ↗

VERIFIED CONSTRUCTIONpairwise overlap heat of 12 directions; brighter is closer

Getting a feel for it

What it defends against

Redundant measurements exist so the signal survives losing a few. A full-spark frame demands that any d survivors span the space; this problem goes further and asks how far the worst set of survivors is from degenerate — the concern of sparse signal processing, erasure-robust transmission and phase retrieval.

Where the frontier is

Existence and constructions of full-spark frames are well studied, but maximizing the worst subset volume at fixed (n, d) has no published table of optima. Every sub-problem is open.

Source

Current leader

0.000000000083

the worst subset's normalized volume

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

ANSWER FORMAT

How to write your answer

Each measurement direction is a nonzero vector: d coordinates written as decimal strings in [-1, 1], at most nine decimal places.

Submit vectors: exactly n rows of d decimal-string coordinates in [-1, 1], each row a nonzero vector.

The current leader's answer

{
  "vectors": [
    [
      "-0.793663828",
      "-0.51443606",
      "-0.081524838",
      "-0.207187479"
    ],
    [
      "-0.792663828",
      "-0.514436060",
      "-0.081524838",
      "-0.207187479"
    ],
    [
      "0.871013056",
      "0.951242353",
      "0.08148684",
      "-0.736974759"
    ],
    [
      "0.775579422",
      "0.735800003",
      "0.541963046",
      "-0.386661121"
    ],
    [
      "-0.357850419",
      "0.924865979",
      "-0.055903268",
      "0.422751127"
    ],
    [
      "0.5547175",
      "0.13890794",
      "-0.898538493",
      "-0.810121174"
    ],
    [
      "-0.618932815",
      "-0.43514097",
      "0.945374578",
      "0.667690955"
    ],
    [
      "-0.730584377",
      "-0.078135778",
      "-0.68906106",
      "0.750884052"
    ],
    [
      "0.881627397",
      "0.332237473",
      "-0.313697535",
      "0.406503276"
    ],
    [
      "-0.827185464",
      "-0.171010341",
      "0.464387418",
      "0.784027342"
    ],
    [
      "0.311301307",
      "0.095519272",
      "0.541259318",
      "0.127355196"
    ],
    [
      "0.40812327",
      "-0.17830549",
      "-0.818953272",
      "-0.756298426"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 12,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      "-0.793663828",
      "-0.51443606",
      "-0.081524838",
      "-0.207187479"
    ],
    [
      "-0.792663828",
      "-0.514436060",
      "-0.081524838",
      "-0.207187479"
    ],
    [
      "0.871013056",
      "0.951242353",
      "0.08148684",
      "-0.736974759"
    ],
    [
      "0.775579422",
      "0.735800003",
      "0.541963046",
      "-0.386661121"
    ],
    [
      "-0.357850419",
      "0.924865979",
      "-0.055903268",
      "0.422751127"
    ],
    [
      "0.5547175",
      "0.13890794",
      "-0.898538493",
      "-0.810121174"
    ],
    [
      "-0.618932815",
      "-0.43514097",
      "0.945374578",
      "0.667690955"
    ],
    [
      "-0.730584377",
      "-0.078135778",
      "-0.68906106",
      "0.750884052"
    ],
    [
      "0.881627397",
      "0.332237473",
      "-0.313697535",
      "0.406503276"
    ],
    [
      "-0.827185464",
      "-0.171010341",
      "0.464387418",
      "0.784027342"
    ],
    [
      "0.311301307",
      "0.095519272",
      "0.541259318",
      "0.127355196"
    ],
    [
      "0.40812327",
      "-0.17830549",
      "-0.818953272",
      "-0.756298426"
    ]
  ]
}

Submit vectors: exactly n rows of d decimal-string coordinates in [-1, 1], each row a nonzero vector. · Verifier v1.0.0

DISCUSSION

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