Shortest moving distance and entropy of morphological complex

(A,B)---4*4*2---(1,0)(0,1)

Classify with neural network A and B, Give Way A and B All are 4*4 Matrix , among A In all the grids of 1,B There is 1 individual 0. There are only 1 A picture , Every picture repeats continuously to realize network convergence . In convergence error δ Consistent case , Count the average number of iterations . Get the form

	431	432	433	434
δ	The number of iterations n	The number of iterations n	The number of iterations n	The number of iterations n
5.00E-04	49664.99	49837.2	50115.58	49468.1
4.00E-04	61174.29	60516.34	60769.59	60547.6
3.00E-04	79569.04	78832.1	79287.35	80434.3
2.00E-04	116011.4	116304.9	115951.6	115654.2
1.00E-04	224611.3	223524.3	225130.4	225118.7

Put this 4 Draw lines together

this 4 The lines are highly coincident , It reflects the symmetrical relationship .

(A,B)---4*4*2---(1,0)(0,1)

Do the second group again ,A It's the same ,B There are two 0, Statistics 4 In this case , The number of iterations obtained is as follows

	421	422	423	424
δ	The number of iterations n	The number of iterations n	The number of iterations n	The number of iterations n
5.00E-04	43419.55	43768.71	44255.1	44394.1
4.00E-04	53934.45	53668.25	54321.85	54310.12
3.00E-04	70134.04	70128.54	70742.93	71375.41
2.00E-04	103634.2	104216.6	103698.3	103876
1.00E-04	200992.7	201399.3	201920.7	201201.3

Due to the existence of symmetry , this 4 The lines are also highly coincident

But compare 43 The data and 42 The data will find

43	1.00E-04	224611.3	223524.3	225130.4	225118.7
42	1.00E-04	200992.7	201399.3	201920.7	201201.3

43 The data of is greater than 42 The data of , Consider the assumption that the number of iterations is inversely proportional to entropy , This shows that 42 Entropy of is greater than 43 The entropy of

For example, classify those involved A and B The numbers in are regarded as components A and B The particles of the universe , And the process of classification is to make A and B The particles in exchange positions with each other , The so-called entropy is the distance of particle motion . For example, for 431, obviously B Medium 3 Particles don't need to move , because A Medium （1,0）,（0,1）,（1,1）3 The two positions are the same 3 A particle .

And the only thing that needs to be moved is A Medium （0,0） The particle , And the shortest moving method of this particle has two kinds of the same probability , Namely （0,0）→（1,0） and （0,0）→（0,1）. The distance of each movement is 1, therefore 431 The corresponding particle moving distance is 1.

And yes 421, Similarly, just move A in （0,1） and （1,1）, Two particles . In the same way , The least expensive way to move these two particles is （0,1）→（0,0）, and （1,1）→（1,0）, Every moving distance is 1, So the total distance is 2, So moving distance and 421>431, therefore 431 The number of iterations of is greater than 421.

The third group of experiments was done in the same way ,B There are only 1 individual 1. Give Way A And B classification

The obtained iteration times table

	411	412	413	414
δ	The number of iterations n	The number of iterations n	The number of iterations n	The number of iterations n
5.00E-04	41909.38	41784.2	42049.96	41667.75
4.00E-04	51967.94	52190.18	51324.76	51861.47
3.00E-04	67093.26	67083.14	67088.89	66872.62
2.00E-04	98633.1	98606.23	101325.4	99056.68
1.00E-04	192559.2	193504.3	194800.3	192974.1

Also this 4 The lines are highly coincident

Compare the number of iterations when the convergence error is consistent

43	1.00E-04	224611.3	223524.3	225130.4	225118.7
42	1.00E-04	200992.7	201399.3	201920.7	201201.3
41	1.00E-04	192559.2	193504.3	194800.3	192974.1

obviously 41 The series is the smallest ,41<42<43.

Calculate with the same method that particles move with each other ,A and B classification , Only A Medium 3 Particles need to be directed B The position of the only particle in moves , And the moving distance is 1+1+1.414=3.414. And this value is greater than 42 Series of 2, Greater than 41 Series of 1. therefore 41 The entropy of the series is the largest . The number of iterations is the smallest .

Therefore, the total moving distance 41>42>43, The number of iterations 41<42<43.