Editing Matrix norm (section)

==={{math|''L''<sub>2,1</sub>}} and {{math|''L<sub>p,q</sub>''}} norms===

Let <math>(a_1, \ldots, a_n) </math> be the dimension {{mvar|m}} columns of matrix <math>A</math>. From the original definition, the matrix <math> A </math> presents {{mvar|n}} data points in an {{mvar|m}}-dimensional space. The <math>L_{2,1}</math> norm<ref>{{cite conference | last1=Ding | first1=Chris | last2=Zhou | first2=Ding | last3=He | first3=Xiaofeng | last4=Zha | first4=Hongyuan  |date = June 2006 | title = R1-PCA: Rotational invariant L1-norm principal component analysis for robust subspace factorization | conference = 23rd International Conference on Machine Learning | series=ICML '06 | isbn = 1-59593-383-2 | place = Pittsburgh, PA | pages=281–288 | doi=10.1145/1143844.1143880 | publisher=[[Association for Computing Machinery]] }}</ref> is the sum of the Euclidean norms of the columns of the matrix:

:<math>\| A \|_{2,1} = \sum_{j=1}^n \| a_{j} \|_2 = \sum_{j=1}^n \left( \sum_{i=1}^m |a_{ij}|^2 \right)^{1/2}</math>

The <math>L_{2,1}</math> norm as an error function is more robust, since the error for each data point (a column) is not squared. It is used in [[robust data analysis]] and [[sparse coding]].

For {{nowrap|''p'', ''q'' ≥ 1}}, the <math>L_{2,1}</math> norm can be generalized to the <math>L_{p,q}</math> norm as follows:

:<math>\| A \|_{p,q} =  \left(\sum_{j=1}^n \left( \sum_{i=1}^m |a_{ij}|^p \right)^{\frac{q}{p}}\right)^{\frac{1}{q}}.</math>