1{
  2  "seed": 1468,
  3  "lambda": 1e-08,
  4  "prediction_1_identity_rank": {
  5    "predicted": "ico rank=6; coordinate rank=3",
  6    "observed": {
  7      "ico_rank": 6,
  8      "coordinate_rank": 3,
  9      "ico_sigma_min": 0.8944271909999157,
 10      "coordinate_smallest_singular_value": 0.0
 11    }
 12  },
 13  "prediction_2_rank_preservation": {
 14    "predicted": "ico rank remains 6 for every invertible F; coordinate remains rank 3",
 15    "observed": [
 16      {
 17        "anisotropy": 0.0,
 18        "ico_rank": 6,
 19        "coordinate_rank": 3,
 20        "ico_sigma_min": 0.8944271909999157,
 21        "coordinate_smallest_nonzero_singular_value": 1.0,
 22        "coordinate_has_zero_singular_values": true
 23      },
 24      {
 25        "anisotropy": 0.5,
 26        "ico_rank": 6,
 27        "coordinate_rank": 3,
 28        "ico_sigma_min": 0.5424975142220966,
 29        "coordinate_smallest_nonzero_singular_value": 0.7094545301398459,
 30        "coordinate_has_zero_singular_values": true
 31      },
 32      {
 33        "anisotropy": 1.0,
 34        "ico_rank": 6,
 35        "coordinate_rank": 3,
 36        "ico_sigma_min": 0.32904137519359206,
 37        "coordinate_smallest_nonzero_singular_value": 0.481646139494879,
 38        "coordinate_has_zero_singular_values": true
 39      },
 40      {
 41        "anisotropy": 1.5,
 42        "ico_rank": 6,
 43        "coordinate_rank": 3,
 44        "ico_sigma_min": 0.19957368236892148,
 45        "coordinate_smallest_nonzero_singular_value": 0.22372305543972865,
 46        "coordinate_has_zero_singular_values": true
 47      },
 48      {
 49        "anisotropy": 2.0,
 50        "ico_rank": 6,
 51        "coordinate_rank": 3,
 52        "ico_sigma_min": 0.12104755722850132,
 53        "coordinate_smallest_nonzero_singular_value": 0.5757695402039034,
 54        "coordinate_has_zero_singular_values": true
 55      },
 56      {
 57        "anisotropy": 2.5,
 58        "ico_rank": 6,
 59        "coordinate_rank": 3,
 60        "ico_sigma_min": 0.07341905474240482,
 61        "coordinate_smallest_nonzero_singular_value": 0.946437150034156,
 62        "coordinate_has_zero_singular_values": true
 63      },
 64      {
 65        "anisotropy": 3.0,
 66        "ico_rank": 6,
 67        "coordinate_rank": 3,
 68        "ico_sigma_min": 0.04453090770838472,
 69        "coordinate_smallest_nonzero_singular_value": 1.2625973447598366,
 70        "coordinate_has_zero_singular_values": true
 71      }
 72    ]
 73  },
 74  "prediction_3_noise_conditioning": {
 75    "predicted": "noise error increases with inverse smallest singular value; coordinate is non-identifiable",
 76    "noise_std": 0.001,
 77    "trials": 100,
 78    "observed": [
 79      {
 80        "anisotropy": 0.0,
 81        "icosahedral": {
 82          "relative_error_mean": 0.0012704696432291392,
 83          "relative_error_std": 0.000595841096511093,
 84          "inv_sigma_min_mean": 1.1180339887498953,
 85          "condition_mean": 1.58113883008419
 86        },
 87        "random6": {
 88          "relative_error_mean": 0.010035117205568666,
 89          "relative_error_std": 0.007045079649234555,
 90          "inv_sigma_min_mean": 29.106303243105593,
 91          "condition_mean": 55.40525057140209
 92        },
 93        "coordinate": {
 94          "relative_error_mean": 0.7509031535351902,
 95          "relative_error_std": 0.19237631049438733,
 96          "inv_sigma_min_mean": 1.0,
 97          "condition_mean": 1.0
 98        }
 99      },
100      {
101        "anisotropy": 0.5,
102        "icosahedral": {
103          "relative_error_mean": 0.0016759503168395102,
104          "relative_error_std": 0.0009917374340478293,
105          "inv_sigma_min_mean": 1.84332641861766,
106          "condition_mean": 5.520952752916615
107        },
108        "random6": {
109          "relative_error_mean": 0.011727149276517907,
110          "relative_error_std": 0.008511823816994935,
111          "inv_sigma_min_mean": 34.78187676411992,
112          "condition_mean": 107.1529411243173
113        },
114        "coordinate": {
115          "relative_error_mean": 0.7064669842638811,
116          "relative_error_std": 0.1677456386351106,
117          "inv_sigma_min_mean": 1.4131901704912775,
118          "condition_mean": 3.0640291801418793
119        }
120      },
121      {
122        "anisotropy": 1.0,
123        "icosahedral": {
124          "relative_error_mean": 0.001955061256420879,
125          "relative_error_std": 0.0010576025339922874,
126          "inv_sigma_min_mean": 3.0391314752184244,
127          "condition_mean": 24.606451260891898
128        },
129        "random6": {
130          "relative_error_mean": 0.01953033383886621,
131          "relative_error_std": 0.021525895369767734,
132          "inv_sigma_min_mean": 55.14237410610069,
133          "condition_mean": 451.59089847290375
134        },
135        "coordinate": {
136          "relative_error_mean": 0.6601241757105272,
137          "relative_error_std": 0.21227515513759707,
138          "inv_sigma_min_mean": 1.7746943455139106,
139          "condition_mean": 10.513500543961538
140        }
141      },
142      {
143        "anisotropy": 1.5,
144        "icosahedral": {
145          "relative_error_mean": 0.0035073731163582245,
146          "relative_error_std": 0.0021749712428775505,
147          "inv_sigma_min_mean": 5.01068070764688,
148          "condition_mean": 110.24953183759459
149        },
150        "random6": {
151          "relative_error_mean": 0.03586979694229784,
152          "relative_error_std": 0.04894632423397871,
153          "inv_sigma_min_mean": 91.08716803115443,
154          "condition_mean": 2039.831902718666
155        },
156        "coordinate": {
157          "relative_error_mean": 0.6409480505081112,
158          "relative_error_std": 0.19058423030337004,
159          "inv_sigma_min_mean": 1.9217242575316122,
160          "condition_mean": 31.336629130650948
161        }
162      },
163      {
164        "anisotropy": 2.0,
165        "icosahedral": {
166          "relative_error_mean": 0.0060951171569506605,
167          "relative_error_std": 0.004521106462034513,
168          "inv_sigma_min_mean": 8.261215863384216,
169          "condition_mean": 494.0976831195629
170        },
171        "random6": {
172          "relative_error_mean": 0.06118769093632997,
173          "relative_error_std": 0.07527994978442538,
174          "inv_sigma_min_mean": 148.11066890806288,
175          "condition_mean": 8845.367386430931
176        },
177        "coordinate": {
178          "relative_error_mean": 0.6127052025895047,
179          "relative_error_std": 0.1946151960050764,
180          "inv_sigma_min_mean": 2.014373407204584,
181          "condition_mean": 89.6416797921853
182        }
183      },
184      {
185        "anisotropy": 2.5,
186        "icosahedral": {
187          "relative_error_mean": 0.010416620657133549,
188          "relative_error_std": 0.007280759986288966,
189          "inv_sigma_min_mean": 13.6204423158071,
190          "condition_mean": 2214.3907496289157
191        },
192        "random6": {
193          "relative_error_mean": 0.0983556860757841,
194          "relative_error_std": 0.08012111424667348,
195          "inv_sigma_min_mean": 238.66808383508354,
196          "condition_mean": 36154.15117052386
197        },
198        "coordinate": {
199          "relative_error_mean": 0.6190785925765124,
200          "relative_error_std": 0.1987391274144127,
201          "inv_sigma_min_mean": 2.1607479661814075,
202          "condition_mean": 259.1801471278394
203        }
204      },
205      {
206        "anisotropy": 3.0,
207        "icosahedral": {
208          "relative_error_mean": 0.016115070630707527,
209          "relative_error_std": 0.010436245198162317,
210          "inv_sigma_min_mean": 22.45631296241717,
211          "condition_mean": 9924.210499549661
212        },
213        "random6": {
214          "relative_error_mean": 0.1688605522348797,
215          "relative_error_std": 0.13056222189022024,
216          "inv_sigma_min_mean": 408.0125935163771,
217          "condition_mean": 188053.61435777947
218        },
219        "coordinate": {
220          "relative_error_mean": 0.6012848091403088,
221          "relative_error_std": 0.20733236015966489,
222          "inv_sigma_min_mean": 2.5483234784416027,
223          "condition_mean": 836.6397187169674
224        }
225      }
226    ]
227  },
228  "exact_reconstruction_max_relative_error": {
229    "icosahedral": 1.781274422149397e-09,
230    "random6": 8.47308806937353e-07
231  },
232  "quantitative_predictions": {
233    "P1_identity_rank": {
234      "predicted_icosa_rank": 6,
235      "observed_icosa_rank": 6,
236      "predicted_coordinate_rank": 3,
237      "observed_coordinate_rank": 3
238    },
239    "P2_SL3_rank_invariance": {
240      "predicted_icosa_rank_at_all_strengths": 6,
241      "observed_icosa_ranks": [
242        6,
243        6,
244        6,
245        6,
246        6,
247        6,
248        6
249      ],
250      "predicted_coordinate_rank_at_all_strengths": 3,
251      "observed_coordinate_ranks": [
252        3,
253        3,
254        3,
255        3,
256        3,
257        3,
258        3
259      ]
260    },
261    "P3_noise_amplification": {
262      "prediction": "error rises as inverse smallest singular value rises",
263      "strengths": [
264        0.0,
265        0.5,
266        1.0,
267        1.5,
268        2.0,
269        2.5,
270        3.0
271      ],
272      "ico_error_at_strength_0_and_3": [
273        0.0012704696432291392,
274        0.016115070630707527
275      ],
276      "ico_inverse_sigma_at_strength_0_and_3": [
277        1.1180339887498953,
278        22.45631296241717
279      ],
280      "error_growth_factor": 12.684341350925964,
281      "inverse_sigma_growth_factor": 20.08553692318978
282    }
283  },
284  "summary": {
285    "identity_ico_full_rank": true,
286    "identity_coordinate_deficient": true,
287    "all_ico_swept_full_rank": true,
288    "all_coordinate_swept_deficient": true,
289    "exact_ico_error": 1.781274422149397e-09,
290    "exact_random6_error": 8.47308806937353e-07
291  }
292}