如标题所描述的，数据无损压缩极限从1948年香农提出“信息熵”以来就界定信源编码的理论极限。此次，基于我提出的加权概率模型，我给出可以无穷例举的实验。该实验的目的就是为了实现“无损降熵”和“无损增熵”。凡是学过信息论和编码技术的朋友，听到这个是不是觉得我在天方夜谭？
不急！！我先给出三组数据，每组256个（0-255的随机数字）字节，请针对下面三组数据进行无损压缩：
第一组：3,172,216,20,41,187,55,239,218,71,179,55,215,130,144,0,236,228,150,29,68,94,2,4,208,108,219,77,236,55,15,56,34,113,29,140,207,18,164,20,0,25,0,57,55,80,2,212,244,151,73,68,107,224,121,123,48,251,29,61,149,122,115,124,104,63,61,168,102,8,40,209,183,146,27,15,248,5,204,215,85,251,187,169,61,174,59,182,118,55,57,178,8,56,88,162,20,151,114,125,204,34,24,47,120,54,145,49,173,40,69,24,14,190,104,1,56,0,2,4,203,106,215,39,178,62,238,48,16,219,180,182,150,254,54,192,48,146,192,136,236,211,195,213,53,66,31,211,132,107,82,93,30,231,25,102,137,242,171,192,188,76,158,114,0,0,9,121,129,97,154,109,53,76,91,50,196,223,226,0,45,44,172,56,54,176,65,237,70,112,0,0,255,251,13,22,229,133,121,141,90,6,3,136,231,159,204,126,104,13,143,156,49,174,222,0,31,229,201,142,141,155,94,163,198,128,215,98,250,187,155,219,70,32,107,124,31,70,98,192,14,255,106,2,213,200,187,12,0,122,242,22,192,97,134,54,10,0,0,0
第二组：7,46,42,168,175,245,71,130,223,45,146,35,246,16,38,52,173,51,88,77,167,52,159,151,27,152,28,136,53,212,88,28,55,237,186,182,57,199,175,107,61,165,19,205,7,180,107,136,230,217,18,17,95,82,253,96,96,224,116,35,233,22,226,29,126,211,121,151,124,49,21,10,22,153,1,138,61,43,6,104,114,152,230,58,106,252,11,91,189,16,224,123,196,156,254,160,219,158,213,185,35,59,145,65,221,160,249,180,16,182,185,174,8,253,211,177,57,59,221,87,222,188,52,193,167,12,82,218,228,93,35,118,64,157,112,175,182,213,54,207,254,87,214,187,154,134,7,15,134,218,237,64,63,214,55,23,86,118,180,27,107,82,142,75,20,239,65,217,244,132,58,167,95,138,245,9,223,51,78,192,139,130,63,0,239,64,147,152,103,212,42,232,169,197,211,26,138,85,43,67,1,118,206,242,68,49,38,176,85,133,195,173,242,1,126,87,61,215,137,159,252,185,150,174,137,200,73,189,232,182,99,244,162,98,121,198,241,21,205,150,137,112,141,101,105,113,77,5,228,85,23,139,252,79,63,166,197,152,0,0
第三组：16,168,14,177,241,9,215,81,64,222,107,65,220,124,9,47,41,47,144,197,3,204,109,19,255,143,25,141,166,66,182,162,96,77,38,133,140,84,146,229,8,63,225,225,40,167,124,224,98,97,59,221,215,65,222,236,107,220,3,219,149,124,196,96,91,245,107,136,181,235,192,93,45,121,9,92,24,123,80,153,187,216,190,171,91,39,106,215,232,63,210,32,137,162,146,148,63,232,149,63,141,69,136,102,186,139,19,102,114,116,227,109,36,53,1,219,225,116,242,168,229,68,50,81,33,8,181,229,168,12,14,231,214,70,100,37,66,60,96,6,249,230,87,100,150,30,242,230,12,92,39,217,110,42,11,163,21,53,127,4,152,236,134,131,115,250,196,98,60,209,22,122,117,107,15,66,17,82,144,243,143,46,248,85,152,40,59,203,168,62,167,13,189,129,169,44,157,118,163,122,222,131,23,203,81,194,71,96,21,237,110,150,158,211,244,25,164,74,15,31,186,50,234,117,132,63,158,27,65,255,117,187,21,11,0,251,221,69,102,29,245,125,204,118,21,234,97,186,185,158,54,132,50,181,211,1,82,107,76,0
根据统计，我们很容易得出上述三组数据中1的概率分别为0.462981，0.504327，0.491346。根据信息论，想在这样的随机数中找出规律进行无损压缩基本不可能。
按照信息熵，上述三组数据将无法再压缩（这个结论我觉得不必要我去描述相关的定理了吧）。
但是基于我提出加权概率模型却能将三者全部进行无损降熵后再压缩。降熵后，第一组可无损压缩至32个字节，第二组可无损压缩至64个字节，第三组可无损压缩至128个字节。
看到这里，是不是嗤之以鼻，怎么可能？往往搞研究的意义就是你“认为”不可能！
接下来我再给出三组数据：
第A组：1,0,0,0,1,0,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0,0,0,1,0,0,1,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,1,1,1,1,1,0,0,0,0,1,0,1,1,1,0,1,1,0,1,1,1,1,1,0,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0,0,1,1,0,0,0,0,0,1,0,1,1,1,1,0,0,1,0,0,0,1,1,1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,1,0,1,1,0,1,0,0,1,0,0,0,0,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0,1,0,0,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,0,0,0,0,1,1,0,0,1,0,0,0,1,1,1,0,0
第B组：1,3,2,0,1,0,0,3,0,0,2,3,1,3,0,1,2,2,1,2,1,2,2,3,3,3,1,2,2,2,1,3,3,3,0,2,2,3,3,3,2,1,3,3,3,0,0,1,2,0,2,2,2,0,1,0,0,3,1,1,2,2,2,2,3,2,2,1,2,1,0,0,0,3,0,3,2,1,1,1,0,1,1,2,3,1,3,3,1,0,1,2,2,2,1,0,1,2,1,0,1,2,2,2,1,3,1,0,3,3,3,0,0,3,3,2,2,0,1,3,2,3,2,2,0,3,1,1,1,0,0,0,3,2,3,2,3,0,0,0,0,1,3,2,2,3,0,2,1,2,3,0,3,3,3,0,3,2,3,2,3,0,1,3,1,1,3,2,1,1,2,2,2,1,1,3,2,1,3,3,3,3,1,2,0,2,2,0,3,1,3,2,0,2,0,3,1,2,0,1,3,1,2,2,2,1,0,0,1,0,1,1,0,3,1,2,1,2,0,2,0,0,0,2,2,1,2,0,0,3,3,0,2,1,2,1,2,3,1,0,1,2,1,2,0,2,2,3,2,1,3,3,2,1,2,0
第C组：3,9,6,7,7,13,11,13,6,14,8,4,6,15,4,7,6,3,2,14,1,5,5,13,7,6,6,14,13,7,3,1,4,10,10,0,8,4,7,8,10,6,10,5,0,12,13,12,11,5,4,15,9,5,5,13,13,7,10,8,12,7,5,11,8,15,3,12,2,0,4,14,12,3,11,10,13,0,1,15,8,0,14,7,15,14,10,12,4,3,1,11,1,2,6,7,9,0,9,2,14,8,11,8,3,15,1,12,5,3,8,6,6,9,9,13,10,8,13,2,6,0,2,13,5,15,14,13,13,5,3,10,2,6,4,12,9,0,7,1,0,13,11,11,12,11,12,9,2,11,5,8,10,7,6,8,1,7,6,3,14,5,10,12,7,0,6,5,5,14,3,4,6,6,2,6,1,13,0,3,15,14,11,5,8,15,4,4,11,2,5,3,3,10,5,1,6,15,10,8,15,3,14,10,15,5,5,13,15,6,13,4,9,11,7,1,8,2,5,7,13,8,5,5,3,10,13,0,5,11,7,15,15,5,15,13,3,3,10,10,13,15,10,3,8,8,0,4,2,6,1,11,13,1,11,15
显然，A组按照8位合成一个字节，于是只需要32个字节；B组按照2位合成一个字节，需要64个字节；C组按照4位合成一个字节，需要128个字节。但是A组与第一组存在什么关系呢？
因为我通过加权概率模型将A组无损增熵得到的第一组数据，于是第一组数据可以被无损降熵为A。这也就是我说我可以将第一组数据无损压缩到32个字节，即无损压缩8倍。同理，也就说明了第二组和第三组均能无损压缩。
我可以明确的告诉大家，无论怎么穷举不同长度的数据，只要能通过无损增熵得到的随机数据，均能被无损降熵。下面给出一部分我的实验代码：
-
#include <stdio.h>
-
#include <stdlib.h>
-
#include <windows.h>
-
#include <time.h>
-
#include "WJLEntropyTransformation.h"
-
#ifdef WIN32
-
#define inline __inline
-
#endif // WIN32
-
// 每个字节中符号1的个数，方便统计
-
unsigned
char CntOfOneSymboltemp[
256]=
-
{
-
0x00,
0x01,
0x01,
0x02,
0x01,
0x02,
0x02,
0x03,
0x01,
0x02,
0x02,
0x03,
0x02,
0x03,
0x03,
0x04,
-
0x01,
0x02,
0x02,
0x03,
0x02,
0x03,
0x03,
0x04,
0x02,
0x03,
0x03,
0x04,
0x03,
0x04,
0x04,
0x05,
-
0x01,
0x02,
0x02,
0x03,
0x02,
0x03,
0x03,
0x04,
0x02,
0x03,
0x03,
0x04,
0x03,
0x04,
0x04,
0x05,
-
0x02,
0x03,
0x03,
0x04,
0x03,
0x04,
0x04,
0x05,
0x03,
0x04,
0x04,
0x05,
0x04,
0x05,
0x05,
0x06,
-
0x01,
0x02,
0x02,
0x03,
0x02,
0x03,
0x03,
0x04,
0x02,
0x03,
0x03,
0x04,
0x03,
0x04,
0x04,
0x05,
-
0x02,
0x03,
0x03,
0x04,
0x03,
0x04,
0x04,
0x05,
0x03,
0x04,
0x04,
0x05,
0x04,
0x05,
0x05,
0x06,
-
0x02,
0x03,
0x03,
0x04,
0x03,
0x04,
0x04,
0x05,
0x03,
0x04,
0x04,
0x05,
0x04,
0x05,
0x05,
0x06,
-
0x03,
0x04,
0x04,
0x05,
0x04,
0x05,
0x05,
0x06,
0x04,
0x05,
0x05,
0x06,
0x05,
0x06,
0x06,
0x07,
-
0x01,
0x02,
0x02,
0x03,
0x02,
0x03,
0x03,
0x04,
0x02,
0x03,
0x03,
0x04,
0x03,
0x04,
0x04,
0x05,
-
0x02,
0x03,
0x03,
0x04,
0x03,
0x04,
0x04,
0x05,
0x03,
0x04,
0x04,
0x05,
0x04,
0x05,
0x05,
0x06,
-
0x02,
0x03,
0x03,
0x04,
0x03,
0x04,
0x04,
0x05,
0x03,
0x04,
0x04,
0x05,
0x04,
0x05,
0x05,
0x06,
-
0x03,
0x04,
0x04,
0x05,
0x04,
0x05,
0x05,
0x06,
0x04,
0x05,
0x05,
0x06,
0x05,
0x06,
0x06,
0x07,
-
0x02,
0x03,
0x03,
0x04,
0x03,
0x04,
0x04,
0x05,
0x03,
0x04,
0x04,
0x05,
0x04,
0x05,
0x05,
0x06,
-
0x03,
0x04,
0x04,
0x05,
0x04,
0x05,
0x05,
0x06,
0x04,
0x05,
0x05,
0x06,
0x05,
0x06,
0x06,
0x07,
-
0x03,
0x04,
0x04,
0x05,
0x04,
0x05,
0x05,
0x06,
0x04,
0x05,
0x05,
0x06,
0x05,
0x06,
0x06,
0x07,
-
0x04,
0x05,
0x05,
0x06,
0x05,
0x06,
0x06,
0x07,
0x05,
0x06,
0x06,
0x07,
0x06,
0x07,
0x07,
0x08
-
};
-
-
int main(){
-
long cont1 =
0, cont2 =
0;
-
unsigned
int enoutlength =
512, length =
512;
-
unsigned
int deoutlength =
512;
-
int i =
0;
-
unsigned
char *TestArray;
-
unsigned
char *enoutArray;
-
unsigned
char *deoutArray;
-
-
double ProbabilityofOne =
0.0;
-
// random datas
-
printf(
"the length is %d for random datas:\n",length);
-
TestArray = (
unsigned
char *)
malloc(
sizeof(
unsigned
char) * length);
-
srand(time(
0));
-
for(i =
0; i < length; ++i) {
-
TestArray[i] = rand() %
4;
// 这个地方可以修改成rand() % 2；rand() % 3；rand() % 16等等
-
printf(
"%d,",(
unsigned
char)TestArray[i]);
-
}
-
printf(
"\n");
-
printf(
"------------------------------------test Increase Entropy ------------------------------------\n");
-
enoutArray = WJLLosslessIncreaseEntropyCoding(TestArray, length, &enoutlength, &ProbabilityofOne);
-
for(i =
0; i < enoutlength; ++i) {
-
printf(
"%d,",(
unsigned
char)enoutArray[i]);
-
cont1 += CntOfOneSymboltemp[enoutArray[i]];
-
}
-
printf(
"\n");
-
printf(
"原始数据中符号1的概率：%f,",ProbabilityofOne);
-
printf(
"\n");
-
printf(
"增熵编码结果中符号1的概率：%f,",(
double)cont1/ (enoutlength *
8.0));
-
printf(
"\n");
-
printf(
"\n");
-
printf(
"-------------------------------------test Drop Entropy ---------------------------------------\n");
-
deoutArray = WJLLosslessDropEntropyCoding(enoutArray, enoutlength, &deoutlength, &ProbabilityofOne);
-
for(i =
0; i < deoutlength; ++i) {
-
printf(
"%d,",(
unsigned
char)deoutArray[i]);
-
if(deoutArray[i] != TestArray[i]){
-
cont2 ++;
-
}
-
}
-
printf(
"\n差异个数为：%d\n",cont2);
-
printf(
"\n");
-
printf(
"\n");
-
system(
"pause");
-
return
0;
-
}
接下来，我再给几张控制台运行的图：
为了证明这一点，我又设计了实验，由随机函数生成的随机数是否能进行无损降熵。实验得出强制降熵是可以实现，降熵后再增熵时出现部分数值错误，而且降熵的越严重，增熵出现的错误也就越多。
同时，也会出现非常小的概率能使得当前的随机数能被无损降熵。下面是512个随机数形成的一个实验：
66,67,227,79,149,215,197,12,114,61,210,103,32,58,11,103,88,44,167,126,145,64,110,98,129,229,99,24,224,129,165,81,111,98,151,250,57,247,63,15,24,64,236,176,124,131,28,118,104,65,156,7,189,104,241,232,187,202,247,147,197,206,200,202,100,101,223,42,143,64,126,3,42,65,76,103,204,47,67,129,197,173,201,40,96,109,29,180,101,161,245,17,226,178,102,18,43,53,3,7,32,27,73,145,174,39,59,181,153,166,73,48,120,90,118,10,1,183,186,110,134,82,165,185,192,150,70,209,204,184,75,184,115,238,105,97,174,220,255,194,105,79,246,42,136,46,235,213,40,174,144,190,39,196,79,180,230,224,49,176,77,216,255,102,16,34,167,26,49,71,224,181,199,148,17,24,253,19,111,176,93,186,102,74,79,225,59,41,220,250,239,86,184,124,102,56,128,32,201,100,64,80,40,182,56,220,100,161,224,240,76,195,41,67,6,189,6,144,177,65,43,74,72,109,20,140,201,87,74,79,151,233,226,223,29,239,70,144,181,110,56,172,200,55,19,177,54,189,84,185,248,36,90,58,4,154,106,239,112,234,246,22,219,78,31,220,9,133,120,22,205,38,13,49,44,52,164,108,191,242,66,68,89,249,241,48,234,135,193,143,163,26,13,223,91,62,255,47,50,163,86,215,116,114,103,102,192,225,98,220,105,6,215,26,28,232,121,150,195,220,33,83,170,15,22,17,224,95,139,193,225,111,105,59,113,250,79,51,203,103,215,105,251,158,28,33,137,27,121,211,87,64,147,35,204,240,153,19,50,89,202,120,93,17,56,171,142,101,204,128,151,238,139,124,63,101,23,66,231,183,121,79,110,92,29,231,83,229,30,79,25,213,197,61,240,178,76,141,145,92,231,53,106,157,5,123,251,0,210,102,128,231,1,114,208,216,202,135,133,24,46,44,93,123,130,249,224,122,43,74,67,134,167,59,141,156,205,208,248,26,85,115,154,209,154,164,68,91,159,236,171,174,80,18,197,242,11,146,118,144,95,75,108,140,20,120,72,20,107,225,254,89,166,83,126,157,203,77,148,141,166,252,12,210,13,105,27,227,104,242,250,29,112,247,212,21,54,213,226,45,103,24,213,101,207,236,116,116,29,213,134,24
将符号1的概率强制设定为0.6进行降熵编码，得到降熵后的数据如下：
102,200,233,4,247,255,255,207,159,139,255,252,64,211,255,57,253,126,255,255,252,63,253,196,247,95,255,255,239,244,58,222,41,135,240,175,40,212,231,255,233,243,175,155,22,254,54,117,191,189,121,43,247,84,80,255,255,207,182,180,223,60,220,255,53,140,55,170,150,243,230,237,175,143,173,142,49,191,252,24,62,244,143,247,255,255,117,38,22,122,103,182,75,154,111,225,209,252,221,5,19,243,191,254,255,59,255,235,253,255,204,38,247,63,94,209,203,109,0,237,250,252,191,238,33,223,197,203,121,153,123,254,223,217,119,90,79,237,175,139,239,252,97,217,252,126,85,120,83,184,115,182,245,167,237,231,255,216,63,226,70,84,67,247,191,135,217,139,117,191,122,90,140,123,127,246,165,221,183,120,110,234,255,248,215,227,239,255,255,251,151,107,217,245,242,43,189,183,95,255,250,253,188,44,255,255,255,98,152,171,175,59,240,254,135,120,104,225,158,241,75,107,116,126,187,201,153,35,238,105,252,222,102,230,237,191,255,255,223,240,227,247,87,127,255,255,223,251,120,185,20,165,140,98,239,255,253,48,239,40,189,185,240,255,255,255,45,244,247,127,223,111,255,214,119,255,185,141,255,252,106,254,239,198,222,111,177,241,184,187,253,186,247,109,155,213,151,142,213,222,107,149,14,59,177,187,252,145,255,235,213,191,237,255,255,154,254,219,19,255,226,255,253,235,245,123,131,207,79,224,233,173,212,222,113,221,111,255,255,240,71,118,91,190,193,52,255,246,239,15,244,174,167,192,7,238,130,255,247,255,255,255,253,208,125,68,182,201,234,255,243,116,49,255,255,234,255,125,203,101,3,219,179,243,239,255,248,107,197,31,245,219,243,69,113,242,223,50,67,108,62,225,215,214,255,253,174,87,160,249,88,254,254,243,231,255,227,239,255,105,166,15,157,204,63,127,71,172,200,184,255,255,255,255,255,255,145,175,117,46,254,21,79,255,103,245,40,61,236,223,251,29,30,95,221,207,84,94,103,103,159,127,255,255,255,255,232,47,114,206,11,117,5,255,254,218,122,116,251,255,161,111,68,155,250,84,167,70,238,93,159,254,95,255,255,255,255,255,181,191,223,55,14,95,189,79,191,255,199,172,255,242
经过统计，实际降熵后符号1的概率为0.680176，不用说也知道降熵成功了。增熵时却得到如下的数据：
66,67,227,79,149,212***,165***,12,114,61,139***,103,32,58,11,101***,87***,43***,90***,126,134***,64,110,98,129,157***,96***,24,224,129,165,81,111,98,151,250,57,218***,63,15,24,64,236,176,124,131,28,118,104,65,156,7,189,95***,241,232,187,202,247,147,197,206,200,202,100,101,223,42,143,64,126,3,42,65,70***,103,204,47,67,128***,197,160***,201,40,96,109,29,180,101,161,245,17,226,177***,102,18,43,53,2***,231***,32,27,42***,145,174,39,59,181,153,165***,73,48,120,90,118,10,1,183,186,110,134,82,165,185,192,150,70,209,204,184,75,184,115,238,105,96***,174,220,255,194,105,79,246,41***,136,46,235,213,40,174,143***,190,39,196,79,180,230,224,49,176,77,216,255,101***,16,34,167,26,48***,71,224,181,199,148,17,24,207***,19,111,176,68***,169***,102,74,79,225,59,41,220,250,239,66***,184,124,102,56,127***,5***,201,100,64,80,40,182,55***,220,100,161,224,240,76,195,41,67,6,189,6,144,177,65,43,74,72,109,20,140,201,49***,3***,76***,150***,232***,226,222***,253***,219***,70,144,181,110,56,172,200,55,19,113***,54,189,84,185,248,36,90,45***,222***,154,106,239,110***,234,246,22,219,73***,31,219***,248***,133,120,22,205,38,13,49,44,52,164,108,191,242,66,68,89,249,241,48,234,135,193,143,163,26,13,223,91,61***,255,47,50,163,52***,215,116,114,103,85***,192,218***,98,220,105,6,215,26,28,232,121,150,195,220,33,83,125***,192***,22,17,224,95,139,193,225,102***,105,59,113,250,79,51,203,103,215,105,249***,155***,12***,245***,137,27,121,211,87,64,147,35,204,240,152***,242***,50,89,202,120,93,17,56,171,142,101,158***,128,151,238,137***,124,63,101,23,66,231,183,121,79,110,91***,29,231,83,211***,30,79,25,213,197,61,239***,178,76,135***,145,91***,231,53,106,157,5,123,251,0,210,102,128,230***,232***,26***,201***,167***,202,135,133,24,46,44,93,119***,130,249,224,122,43,74,67,134,167,57***,141,156,205,208,248,26,85,112***,100***,201***,154,164,68,91,159,236,171,174,80,18,197,242,11,145***,118,144,95,75,108,140,20,120,72,20,107,212***,254,27***,158***,2***,124***,157,202***,77,148,141,166,252,12,182***,13,105,24***,226***,227***,212***,50***
其中有90个字节在增熵时发生错误（如带***的数值是错误的）。
接下来，我说为什么这个实验可以撼动“信息论”。至少我能找到无穷种随机数据均能被无损降熵，那么就说明“信息论”的理论存在不完整性。
本次只是记录了我的实验发现，我会提供C的lib库供大家测试。
在我看来，这也许是一个新的研究方向，所以先把实验结果发不出来。
转载：https://blog.csdn.net/wjlxueshu/article/details/112607768