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最鲁棒的冗余测量方向 · n = 14

在 R^d 中选择 n 个非零向量。对每个大小为 d 的子集，取以这些向量为列的矩阵的行列式平方除以各向量范数平方之积；最大化所有子集中这个归一化体积的最小值。为零意味着某 d 个幸存测量无法恢复整个空间。

子题d = 5, n = 14

目标最大化最坏子集的归一化体积

严格定义

容器d 维实空间 R^d；答案是 n 个测量方向
提交恰好 n 个非零向量，每个 d 个坐标
目标最大化 min det(V_S)²/Π|vᵢ|²，对全部 C(n,d) 个子集取最小；行列式与范数全是有理数，比较交叉相乘
措辞归一化体积是与数值稳定性一致的鲁棒性代理，不等同于所有噪声模型下的最优重建误差；本题只声称 maximin volume
计分纪录是 floor(最小归一化体积 · 10¹⁸)，向不利于提交者的方向取整；页面显示该体积，向下取整到第 12 位小数

挑战这个纪录 ↗

已验证构造14 个方向的两两重合度热图，越亮越接近

帮助理解

它防的是什么

冗余测量的意义是坏掉几个也能恢复信号。full-spark frame 要求任何 d 个幸存向量都张成全空间；这里更进一步，问最坏的那组幸存者离退化有多远，这出现在稀疏信号处理、抗擦除传输与相位恢复里。

前沿在哪里

full-spark frame 的存在性与构造有成熟文献，但固定 (n, d) 下最大化最坏子集体积没有已发表的最优表。每个子题都开放。

查看来源

当前第一名

0

最坏子集的归一化体积

纪录保持者创始基准

解题方式人工

挑战这个纪录 ↗

ANSWER FORMAT

答案怎么写

每个测量方向是一个非零向量：d 个坐标写成 [-1, 1] 内的十进制字符串，最多九位小数。

提交 vectors：恰好 n 行，每行 d 个 [-1, 1] 内的十进制字符串坐标，每行是一个非零向量。

当前第一名的答案

{
  "vectors": [
    [
      "-0.793663828",
      "-0.51443606",
      "-0.081524838",
      "-0.207187479",
      "0.112075371"
    ],
    [
      "-0.792663828",
      "-0.514436060",
      "-0.081524838",
      "-0.207187479",
      "0.112075371"
    ],
    [
      "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",
      "0.495970837",
      "0.619861665"
    ],
    [
      "-0.671084819",
      "-0.743768305",
      "-0.719986338",
      "0.713555531",
      "0.848777345"
    ],
    [
      "-0.514231321",
      "0.392741697",
      "0.653635797",
      "0.301370616",
      "-0.939891107"
    ],
    [
      "-0.981953745",
      "-0.673711428",
      "-0.224169119",
      "0.434560621",
      "0.195397592"
    ],
    [
      "0.893262348",
      "0.637760527",
      "0.612948301",
      "0.883889483",
      "0.773282469"
    ]
  ]
}

提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 14,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      "-0.793663828",
      "-0.51443606",
      "-0.081524838",
      "-0.207187479",
      "0.112075371"
    ],
    [
      "-0.792663828",
      "-0.514436060",
      "-0.081524838",
      "-0.207187479",
      "0.112075371"
    ],
    [
      "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",
      "0.495970837",
      "0.619861665"
    ],
    [
      "-0.671084819",
      "-0.743768305",
      "-0.719986338",
      "0.713555531",
      "0.848777345"
    ],
    [
      "-0.514231321",
      "0.392741697",
      "0.653635797",
      "0.301370616",
      "-0.939891107"
    ],
    [
      "-0.981953745",
      "-0.673711428",
      "-0.224169119",
      "0.434560621",
      "0.195397592"
    ],
    [
      "0.893262348",
      "0.637760527",
      "0.612948301",
      "0.883889483",
      "0.773282469"
    ]
  ]
}

提交 vectors：恰好 n 行，每行 d 个 [-1, 1] 内的十进制字符串坐标，每行是一个非零向量。 · 验证器 v1.0.0

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