{ "seed": 1234, "math_exact": { "same": 1.000088900582341e-12, "orthogonal": 1.0, "antipodal": 1.999999999999 }, "noise_scaling": [ { "sigma": 0.01, "observed": 0.0006986228358126927, "predicted": 0.0007000000000000001, "ratio": 0.9980326225895608 }, { "sigma": 0.03, "observed": 0.006276764365943627, "predicted": 0.006299999999999999, "ratio": 0.9963118041180361 }, { "sigma": 0.06, "observed": 0.02462667338052535, "predicted": 0.025199999999999997, "ratio": 0.977248943671641 }, { "sigma": 0.1, "observed": 0.06612041432046092, "predicted": 0.07, "ratio": 0.944577347435156 }, { "sigma": 0.18, "observed": 0.19009649754592634, "predicted": 0.2268, "ratio": 0.8381679785975588 }, { "sigma": 0.3, "observed": 0.4062411889336691, "predicted": 0.63, "ratio": 0.644827284021697 } ], "density_sweep": [ { "sigma": 0.01, "within_distance": 0.0007520450488480197, "between_distance": 1.0050393620475382, "between_minus_within": 1.0042873169986901, "cluster_purity": 1.0, "mean_core": 0.0004516606447072108 }, { "sigma": 0.05, "within_distance": 0.02045960371044641, "between_distance": 1.011824199526698, "between_minus_within": 0.9913645958162517, "cluster_purity": 1.0, "mean_core": 0.013977510386516345 }, { "sigma": 0.1, "within_distance": 0.06391457595483079, "between_distance": 0.8807575499937699, "between_minus_within": 0.8168429740389391, "cluster_purity": 1.0, "mean_core": 0.0410638974190513 }, { "sigma": 0.2, "within_distance": 0.21223261844670432, "between_distance": 1.0524008180311804, "between_minus_within": 0.8401681995844761, "cluster_purity": 1.0, "mean_core": 0.1612017055260236 }, { "sigma": 0.35, "within_distance": 0.4622438689769046, "between_distance": 0.8385216512010965, "between_minus_within": 0.37627778222419184, "cluster_purity": 1.0, "mean_core": 0.3086342844447999 }, { "sigma": 0.6, "within_distance": 0.7013901853588502, "between_distance": 0.8659499411151649, "between_minus_within": 0.16455975575631465, "cluster_purity": 1.0, "mean_core": 0.4020908234366968 } ], "training": { "angles": [ 0.0, 0.1, 0.18, 0.25, 1.3, 1.4, 1.48, 1.58 ], "discovered_groups": [ 0, 0, 0, 0, 1, 1, 1, 1 ], "probe_core": [ 0.01615630721187855, 0.0049958347219741794, 0.00319829369738045, 0.011228922063957647, 0.016156307211878662, 0.0049958347219741794, 0.0049958347219742905, 0.016156307211878662 ], "shared_population_loss": 0.3723075967863627, "discovered_population_loss": 0.00958122769370327, "oracle_population_loss": 0.00958122769370327, "conflict_sweep": [ { "angle": 0, "shared_loss": 0.0, "partition_loss": 0.0, "observed_gain": 0.0, "predicted_gain": 0.0 }, { "angle": 0.2, "shared_loss": 0.009966711079379185, "partition_loss": 0.0, "observed_gain": 0.009966711079379185, "predicted_gain": 0.009966711079379187 }, { "angle": 0.4, "shared_loss": 0.03946950299855746, "partition_loss": 0.0, "observed_gain": 0.03946950299855746, "predicted_gain": 0.03946950299855745 }, { "angle": 0.6, "shared_loss": 0.08733219254516086, "partition_loss": 0.0, "observed_gain": 0.08733219254516086, "predicted_gain": 0.08733219254516084 }, { "angle": 0.8, "shared_loss": 0.15164664532641736, "partition_loss": 1.3877787807814457e-17, "observed_gain": 0.15164664532641736, "predicted_gain": 0.1516466453264173 }, { "angle": 1.0, "shared_loss": 0.22984884706593012, "partition_loss": 0.0, "observed_gain": 0.22984884706593012, "predicted_gain": 0.22984884706593012 }, { "angle": 1.2, "shared_loss": 0.31882112276166324, "partition_loss": 0.0, "observed_gain": 0.31882112276166324, "predicted_gain": 0.3188211227616632 }, { "angle": 1.5707963267948966, "shared_loss": 0.5, "partition_loss": 0.0, "observed_gain": 0.5, "predicted_gain": 0.49999999999999994 } ] }, "predictions": { "P1": "cosine distance is 0/1/2 for same/orthogonal/antipodal normalized embeddings", "P2": "for d=8 and small sigma, E[distance] ~= 7 sigma^2", "P3": "for two unit linear tasks at angle theta, partition gain per task = (1-|cos(theta)|)/2" } }