Reference: http: //blog.csdn.net/susanzhang1231/article/details/52127011

Reprinted August 5, 2016 10:54:35

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Just ended, for any errors please exhibitions. Reprinted from http://www.zhihu.com/question/20473040/answer/102907063

From the perspective of functions can, to understand the geometry of the matrix norm.

We all know that there is often a function of the geometry of the correspondence between this well imagine, especially in the space below the three-dimensional, geometric image is a mathematical function of generalization, but is highly geometric image visualization functions, such as a function spatial pattern corresponding to a plurality of geometric points.
But beyond function and geometric three-dimensional space, it is difficult to obtain a good imagination, so there is the concept of mapping, mapping of expression is a collection into another collection through a relationship. Usually mapping math book is the first to say, and then discuss the function, which is a special case because the function is mapped.
In order to better express this relationship mathematically mapping, (here especially linear relationship) so it introduced a matrix. Here the matrix is characterized by a linear relationship between the spatial mapping. And to represent the above mentioned map collection by this vector, we usually refer to the group, this is a collection of the most general relations. Thus, we can understand that a set (vector), by means of a mapping relationship (matrix), to give a further set (another vector).

Then the norm of the vector indicates that the original size of the collection.

Norm matrix represents a measure of the size of the change process.

Briefly: 0 norm represents the number of non-zero elements (i.e., its sparsity) vector. 1 norm expressed as absolute value sum. The norm is fingerprints.

Vector norm

1- norm:

$||x||_1 = \sum_{i=1}^N|x_i|$ , I.e. the sum of absolute vector elements, matlab calling function norm (x, 1).

2-norm:

$||\textbf{x}||_2 =\sqrt{\sum_{i=1}^Nx_i^2}$ , Euclid norm (Euclidean norm, the vector used in the calculation length), i.e., the square of the absolute value of the vector elements and re-evolution, matlab calling function norm (x, 2).

$\infty$ - norm: $||\textbf{x}||_\infty = \max_{i}|x_i|$ that all elements of the vector absolute value of the maximum value, matlab calling function norm (x, inf).

$-\infty$ - norm: $||\textbf{x}||_{-\infty}=\min_i|x_i|$

, I.e. the minimum of all the absolute values of the vector elements, matlab calling function norm (x, -inf).

p- norm: $||\textbf{x}||_p = (\sum_{i=1}^N|x_i|^p)^{\frac{1}{p}}$
, i.e., the absolute value of the vector elements and the power p 1 / p power, matlab calling function norm (x, p).

Matrix norm

1- norm: $||A||_1 = \max_j\sum_{i=1}^m|a_{i,j}|$
columns and norm, i.e. maximum column vector and matrix for all absolute values, matlab calls the function norm (A, 1).

2-norm: $||A||_2 = \sqrt{\lambda_1}$ , $\lambda<br/>$ to the $A TA '$ maximum eigenvalue.

, Spectral norm, i.e., the maximum eigenvalue of the matrix square root A'A. matlab calling function norm (x, 2).

$\infty$ - norm: $||A||_\infty = \max_i\sum_{j=1}^N|a_{i,j}|$

, Norm and rows, i.e. the vector sum of the absolute values of the maximum value of all the matrix rows, matlab calls the function norm (A, inf).

F- norm: $||A||_F=\left(\sum_{i=1}^m\sum_{j=1}^n|a_{i,j}|^2\right)^{\frac{1}{2}}$

, The Frobenius norm, i.e., an absolute value of sum of squares of the matrix elements to open square, matlab calls the function norm (A, 'fro').

Nuclear norm: $||A||_* = \sum_{i=1}^{n}\lambda_i, \lambda_i$ the A's singular values .

That singular values of the sum.

Lp norm regularization is used, of which the L2 norm | w | w 2 tends to try to balance the weight value, i.e., the number of non-zero components dense as possible, and L0 and L1 norm norm is inclined w sparse component as possible, i.e. to minimize the number of non-zero components.