matlab中ridge和A\b的区别
给定相同的 A、b 和 L2 正则化参数 beta = 0，为什么 ridge 和 \给出两种不同的解决方案？
b = [ 0
-2
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3 ];
A = [
1 0 0 0
0.750000000000000 0.250000000000000 0 0
0.500000000000000 0.500000000000000 0 0
0.250000000000000 0.750000000000000 0 0
0 1 0 0
0 0.750000000000000 0.250000000000000 0
0 0.500000000000000 0.500000000000000 0
0 0.250000000000000 0.750000000000000 0
0 0 1 0
0 0 0.750000000000000 0.250000000000000
0 0 0.500000000000000 0.500000000000000
0 0 0.250000000000000 0.750000000000000
0 0 0 1
];
>> ridge(b, A, 0,0)
ans =
0.6942
-0.1856
0
-0.0468
>> A \ b
ans =
-0.8604
-3.4188
-2.8970
-3.0343
Given the same A, b and L2 regularization parameter beta = 0, why do ridge and \ give two different solutions?
b = [ 0
-2
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3 ];
A = [
1 0 0 0
0.750000000000000 0.250000000000000 0 0
0.500000000000000 0.500000000000000 0 0
0.250000000000000 0.750000000000000 0 0
0 1 0 0
0 0.750000000000000 0.250000000000000 0
0 0.500000000000000 0.500000000000000 0
0 0.250000000000000 0.750000000000000 0
0 0 1 0
0 0 0.750000000000000 0.250000000000000
0 0 0.500000000000000 0.500000000000000
0 0 0.250000000000000 0.750000000000000
0 0 0 1
];
>> ridge(b, A, 0,0)
ans =
0.6942
-0.1856
0
-0.0468
>> A \ b
ans =
-0.8604
-3.4188
-2.8970
-3.0343
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这是因为，正如您在 文档 中看到的，
ridge使用的算法与mldivide略有不同：因为“经典”伪逆 ((A' *A)^-1 * A）可能会变成对于接近奇异值的 (A' *A)^-1 的小误差敏感，公式修改为 (A' *< em>A - kI)^-1 *A，减少问题的条件。That's because, as you can se in the documentation,
ridgeuses a slightly different algorithm thanmldivide: since the "classical" pseudoinverse ((A' *A)^-1 *A) could become sensible to small errors for (A' *A)^-1 close to singular values, the formula is modified to (A' *A - kI)^-1 *A, reducing the conditioning of the problem.