{
  "finite_difference_gradient_check": {
    "analytic": [
      0.5887907974608669,
      0.47610356349706445,
      0.09578414886923187
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    "finite_difference": [
      0.5887907974511108,
      0.47610356340710425,
      0.09578414866817653
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    "max_abs_error": 2.0105533904413164e-10
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  "paper_witness": {
    "rank": 4,
    "singular_values": [
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      2.083534946053332,
      1.732050807568877,
      1.2377363871144758
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    "selected_minor_determinant": 1.0
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  "candidate_pool": {
    "candidates": 180,
    "global_edges": 30
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  "all_candidates": {
    "rank": 30,
    "logdet_information": 143.97731772336564,
    "mean_pairwise_cosine_abs": 0.08583190536581144,
    "smallest_singular_value": 3.1874984540089324
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  "budget_sweep": {
    "10%": {
      "budget": 18,
      "greedy": {
        "rank": 18,
        "logdet_information": 13.527887707414422,
        "mean_pairwise_cosine_abs": 0.08491808197644612,
        "smallest_singular_value": 3.894886259616979
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      "greedy_rank_trace": [
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        8,
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        10,
        11,
        12,
        13,
        14,
        15,
        16,
        17,
        18
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      "random_mean": {
        "rank": 18.0,
        "logdet_information": -65.42420465003103,
        "mean_pairwise_cosine_abs": 0.08785599575750976,
        "smallest_singular_value": 0.19661288052855125
      },
      "random_std": {
        "rank": 0.0,
        "logdet_information": 7.8883767462736065,
        "mean_pairwise_cosine_abs": 0.01385093711095459,
        "smallest_singular_value": 0.0902454521127866
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      "message_fraction": 0.1
    },
    "25%": {
      "budget": 45,
      "greedy": {
        "rank": 30,
        "logdet_information": 134.63245768859463,
        "mean_pairwise_cosine_abs": 0.09705173859973731,
        "smallest_singular_value": 2.2128802658610316
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      "greedy_rank_trace": [
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        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
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        15,
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        20,
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        25,
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        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30
      ],
      "random_mean": {
        "rank": 29.725,
        "logdet_information": 53.10032875798106,
        "mean_pairwise_cosine_abs": 0.08491663353953435,
        "smallest_singular_value": 0.18510807484639685
      },
      "random_std": {
        "rank": 0.4993746088859544,
        "logdet_information": 10.831666566067572,
        "mean_pairwise_cosine_abs": 0.00556639421153242,
        "smallest_singular_value": 0.17803958099619155
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      "message_fraction": 0.25
    },
    "50%": {
      "budget": 90,
      "greedy": {
        "rank": 30,
        "logdet_information": 141.86048780360787,
        "mean_pairwise_cosine_abs": 0.09416596334657241,
        "smallest_singular_value": 2.9078365987580344
      },
      "greedy_rank_trace": [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15,
        16,
        17,
        18,
        19,
        20,
        21,
        22,
        23,
        24,
        25,
        26,
        27,
        28,
        29,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30,
        30
      ],
      "random_mean": {
        "rank": 30.0,
        "logdet_information": 101.9889799835815,
        "mean_pairwise_cosine_abs": 0.08575823816591169,
        "smallest_singular_value": 1.180208753849413
      },
      "random_std": {
        "rank": 0.0,
        "logdet_information": 5.881708339239661,
        "mean_pairwise_cosine_abs": 0.0025019575771706294,
        "smallest_singular_value": 0.36951616928349934
      },
      "message_fraction": 0.5
    }
  },
  "interpretation": "Greedy selects rank-increasing rows and maximizes regularized global edge information; random is the equal-budget control."
}