Markov Spectral Equivariant Layer / results.json
Mechanism confirmed, baseline not beaten
1{
2 "seed": 1203,
3 "M": 256,
4 "predictions": [
5 "For every L, min(K)>=0 and sum(K)=1, hence induced sup norm and random error ratios are <=1.",
6 "Matched-rank signed truncation has induced sup norm sum(abs(K)); its worst-case ratio equals that value and generally grows with rank.",
7 "Fejer Fourier eigenvalue at integer frequency k is max(1-|k|/L,0), giving linear attenuation to the cutoff and zero beyond it."
8 ],
9 "kernel_sweep": [
10 {
11 "L": 2,
12 "min_fejer_kernel": 2.9292183255114406e-35,
13 "mass_error": 1.1102230246251565e-16,
14 "fejer_inf_norm": 0.9999999999999999,
15 "signed_inf_norm": 1.4360095172636305,
16 "signed_worst_case_ratio": 1.4360095172636305,
17 "fejer_random_ratio": 0.045108277874436886
18 },
19 {
20 "L": 4,
21 "min_fejer_kernel": 5.858436651022879e-35,
22 "mass_error": 2.220446049250313e-16,
23 "fejer_inf_norm": 0.9999999999999998,
24 "signed_inf_norm": 1.7784319428805193,
25 "signed_worst_case_ratio": 1.7784319428805193,
26 "fejer_random_ratio": 0.048089527403431
27 },
28 {
29 "L": 8,
30 "min_fejer_kernel": 1.1716873302045763e-34,
31 "mass_error": 2.220446049250313e-16,
32 "fejer_inf_norm": 0.9999999999999998,
33 "signed_inf_norm": 2.086461979555266,
34 "signed_worst_case_ratio": 2.086461979555266,
35 "fejer_random_ratio": 0.1350374025518305
36 },
37 {
38 "L": 16,
39 "min_fejer_kernel": 1.1811367237062345e-34,
40 "mass_error": 6.661338147750939e-16,
41 "fejer_inf_norm": 0.9999999999999993,
42 "signed_inf_norm": 2.3826342877350895,
43 "signed_worst_case_ratio": 2.3826342877350895,
44 "fejer_random_ratio": 0.20814128441985832
45 },
46 {
47 "L": 32,
48 "min_fejer_kernel": 1.0240906784363769e-34,
49 "mass_error": 2.220446049250313e-16,
50 "fejer_inf_norm": 1.0000000000000002,
51 "signed_inf_norm": 2.656816980720107,
52 "signed_worst_case_ratio": 2.6568169807201065,
53 "fejer_random_ratio": 0.30104351387825673
54 },
55 {
56 "L": 64,
57 "min_fejer_kernel": 6.13608051129234e-35,
58 "mass_error": 0.0,
59 "fejer_inf_norm": 1.0,
60 "signed_inf_norm": 2.797267840356451,
61 "signed_worst_case_ratio": 2.797267840356451,
62 "fejer_random_ratio": 0.4657958293871679
63 }
64 ],
65 "attenuation_sweep": [
66 {
67 "L": 2,
68 "checks": [
69 {
70 "k": 0,
71 "observed": 1.0,
72 "predicted": 1.0,
73 "abs_error": 0.0
74 },
75 {
76 "k": 1,
77 "observed": 0.5,
78 "predicted": 0.5,
79 "abs_error": 0.0
80 },
81 {
82 "k": 1,
83 "observed": 0.5,
84 "predicted": 0.5,
85 "abs_error": 0.0
86 },
87 {
88 "k": 2,
89 "observed": -1.8022814952877887e-17,
90 "predicted": 0.0,
91 "abs_error": 1.8022814952877887e-17
92 },
93 {
94 "k": 5,
95 "observed": 7.01022896296564e-18,
96 "predicted": 0.0,
97 "abs_error": 7.01022896296564e-18
98 }
99 ]
100 },
101 {
102 "L": 4,
103 "checks": [
104 {
105 "k": 0,
106 "observed": 0.9999999999999998,
107 "predicted": 1.0,
108 "abs_error": 2.220446049250313e-16
109 },
110 {
111 "k": 2,
112 "observed": 0.4999999999999999,
113 "predicted": 0.5,
114 "abs_error": 1.1102230246251565e-16
115 },
116 {
117 "k": 3,
118 "observed": 0.25,
119 "predicted": 0.25,
120 "abs_error": 0.0
121 },
122 {
123 "k": 4,
124 "observed": 2.2218053890416e-17,
125 "predicted": 0.0,
126 "abs_error": 2.2218053890416e-17
127 },
128 {
129 "k": 7,
130 "observed": -1.1041124689322516e-17,
131 "predicted": 0.0,
132 "abs_error": 1.1041124689322516e-17
133 }
134 ]
135 },
136 {
137 "L": 8,
138 "checks": [
139 {
140 "k": 0,
141 "observed": 0.9999999999999998,
142 "predicted": 1.0,
143 "abs_error": 2.220446049250313e-16
144 },
145 {
146 "k": 4,
147 "observed": 0.49999999999999994,
148 "predicted": 0.5,
149 "abs_error": 5.551115123125783e-17
150 },
151 {
152 "k": 7,
153 "observed": 0.12500000000000014,
154 "predicted": 0.125,
155 "abs_error": 1.3877787807814457e-16
156 },
157 {
158 "k": 8,
159 "observed": 1.1211633618255831e-16,
160 "predicted": 0.0,
161 "abs_error": 1.1211633618255831e-16
162 },
163 {
164 "k": 11,
165 "observed": 6.938893903907228e-17,
166 "predicted": 0.0,
167 "abs_error": 6.938893903907228e-17
168 }
169 ]
170 },
171 {
172 "L": 16,
173 "checks": [
174 {
175 "k": 0,
176 "observed": 0.9999999999999993,
177 "predicted": 1.0,
178 "abs_error": 6.661338147750939e-16
179 },
180 {
181 "k": 8,
182 "observed": 0.49999999999999994,
183 "predicted": 0.5,
184 "abs_error": 5.551115123125783e-17
185 },
186 {
187 "k": 15,
188 "observed": 0.06250000000000062,
189 "predicted": 0.0625,
190 "abs_error": 6.245004513516506e-16
191 },
192 {
193 "k": 16,
194 "observed": 6.009045720724373e-16,
195 "predicted": 0.0,
196 "abs_error": 6.009045720724373e-16
197 },
198 {
199 "k": 19,
200 "observed": 4.112946560504012e-16,
201 "predicted": 0.0,
202 "abs_error": 4.112946560504012e-16
203 }
204 ]
205 },
206 {
207 "L": 32,
208 "checks": [
209 {
210 "k": 0,
211 "observed": 1.0000000000000002,
212 "predicted": 1.0,
213 "abs_error": 2.220446049250313e-16
214 },
215 {
216 "k": 16,
217 "observed": 0.5000000000000004,
218 "predicted": 0.5,
219 "abs_error": 4.440892098500626e-16
220 },
221 {
222 "k": 31,
223 "observed": 0.03124999999999993,
224 "predicted": 0.03125,
225 "abs_error": 6.938893903907228e-17
226 },
227 {
228 "k": 32,
229 "observed": -1.6148247973969785e-16,
230 "predicted": 0.0,
231 "abs_error": 1.6148247973969785e-16
232 },
233 {
234 "k": 35,
235 "observed": -4.163336342344337e-16,
236 "predicted": 0.0,
237 "abs_error": 4.163336342344337e-16
238 }
239 ]
240 },
241 {
242 "L": 64,
243 "checks": [
244 {
245 "k": 0,
246 "observed": 1.0,
247 "predicted": 1.0,
248 "abs_error": 0.0
249 },
250 {
251 "k": 32,
252 "observed": 0.5000000000000001,
253 "predicted": 0.5,
254 "abs_error": 1.1102230246251565e-16
255 },
256 {
257 "k": 63,
258 "observed": 0.015624999999999917,
259 "predicted": 0.015625,
260 "abs_error": 8.326672684688674e-17
261 },
262 {
263 "k": 64,
264 "observed": -1.6653345369377348e-16,
265 "predicted": 0.0,
266 "abs_error": 1.6653345369377348e-16
267 },
268 {
269 "k": 67,
270 "observed": -3.365363543395006e-16,
271 "predicted": 0.0,
272 "abs_error": 3.365363543395006e-16
273 }
274 ]
275 }
276 ],
277 "equivariance_max_errors": [
278 2.7755575615628914e-17,
279 6.938893903907228e-17,
280 1.1102230246251565e-16,
281 1.8041124150158794e-16,
282 2.220446049250313e-16,
283 2.220446049250313e-16
284 ],
285 "order_violation_max": 0.0,
286 "random_and_stack": [
287 {
288 "L": 2,
289 "mean_fejer_ratio": 0.0583947677047889,
290 "mean_signed_ratio": 0.08859360989821372,
291 "20_layer_signed_norm_ratio": 5.328913391621261
292 },
293 {
294 "L": 4,
295 "mean_fejer_ratio": 0.1044140331177652,
296 "mean_signed_ratio": 0.18709733702224654,
297 "20_layer_signed_norm_ratio": 30.348196518810855
298 },
299 {
300 "L": 8,
301 "mean_fejer_ratio": 0.17177002183166398,
302 "mean_signed_ratio": 0.3207410370611741,
303 "20_layer_signed_norm_ratio": 42.61679913053706
304 },
305 {
306 "L": 16,
307 "mean_fejer_ratio": 0.2841416403889107,
308 "mean_signed_ratio": 0.5253006736515848,
309 "20_layer_signed_norm_ratio": 4.229985449727612
310 },
311 {
312 "L": 32,
313 "mean_fejer_ratio": 0.4279204044860396,
314 "mean_signed_ratio": 0.7889652829318335,
315 "20_layer_signed_norm_ratio": 13.611940109013641
316 },
317 {
318 "L": 64,
319 "mean_fejer_ratio": 0.6265712964133795,
320 "mean_signed_ratio": 1.0815584398602223,
321 "20_layer_signed_norm_ratio": 13.272669290001211
322 }
323 ],
324 "stacked_20_layers": [
325 {
326 "L": 2,
327 "fejer_20_layer_sup_ratio": 0.011489785304145104,
328 "signed_20_layer_sup_ratio": 0.050078718741095575
329 },
330 {
331 "L": 4,
332 "fejer_20_layer_sup_ratio": 0.041079369759704665,
333 "signed_20_layer_sup_ratio": 0.26077283401090773
334 },
335 {
336 "L": 8,
337 "fejer_20_layer_sup_ratio": 0.037491544556016734,
338 "signed_20_layer_sup_ratio": 0.41650962818785814
339 },
340 {
341 "L": 16,
342 "fejer_20_layer_sup_ratio": 0.032817268694412656,
343 "signed_20_layer_sup_ratio": 0.5108483451033049
344 },
345 {
346 "L": 32,
347 "fejer_20_layer_sup_ratio": 0.07191079728031154,
348 "signed_20_layer_sup_ratio": 0.7186433133753423
349 },
350 {
351 "L": 64,
352 "fejer_20_layer_sup_ratio": 0.14606585222262247,
353 "signed_20_layer_sup_ratio": 1.3934100850081501
354 }
355 ],
356 "summary": {
357 "max_fejer_mass_error": 6.661338147750939e-16,
358 "min_fejer_kernel": 2.9292183255114406e-35,
359 "max_equivariance_error": 2.220446049250313e-16,
360 "max_attenuation_error": 6.661338147750939e-16,
361 "signed_norms": [
362 1.4360095172636305,
363 1.7784319428805193,
364 2.086461979555266,
365 2.3826342877350895,
366 2.656816980720107,
367 2.797267840356451
368 ],
369 "fejer_norms": [
370 0.9999999999999999,
371 0.9999999999999998,
372 0.9999999999999998,
373 0.9999999999999993,
374 1.0000000000000002,
375 1.0
376 ]
377 }
378}