Matrix Factorization Task

A Matrix Factorization Task is a decomposition task that requires a matrix factorization structure (that decomposes an [math]\displaystyle{ m \times n }[/math] matrix [math]\displaystyle{ X }[/math] into a product of matrices (in some canonical form) - typically into [math]\displaystyle{ m×k }[/math] and [math]\displaystyle{ k×n }[/math] matrices, where typically [math]\displaystyle{ k \ll n }[/math] and [math]\displaystyle{ k \ll m }[/math]).

Context:
- Input: a Matrix Structure.
- output: a Matrix Factorization Structure.
- It can be solved by a Matrix Decomposition System (that implements a matrix decomposition algorithm).
- It can range from being a Nonnegative Matrix Factorization to being a Positive Matrix Factorization to being ...
- It can range from being an Exact Matrix Decomposition Task to being an Approximate Matrix Factorization Task.
- It can range from being a Regularized Matrix Decomposition Task to being an Non-Regularized Matrix Factorization Task.
- It can range from being a Weighted Matrix Decomposition Task to being an Unweighted Matrix Factorization Task.
- It can range from being a Boolean Matrix Decomposition Task to being an Integer Matrix Decomposition Task to being Real Matrix Decomposition Task.
- It can range from being a Low-Rank Matrix Factorization Task to being a Large-Rank Matrix Factorization Algorithm.
Example(s):
Counter-Example(s):
- Matrix-Matrix Multiplication.
See: Lossy Compression, Adjacency Matrix, Latent Factor Algorithm, Linear Algebra Task.

References

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/matrix_decomposition Retrieved:2015-1-17.
- In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.