Column-Row Factorization (CR)

• categories: Linear algebra, Factorization

Definition

Column-Row Factorization (CR) refers to the representation of a matrix $A\in {\mathbb{R}}^{m\times n}$ as the product of two lower-rank matrices, emphasizing the column and row structures of $A$. Specifically, $A$ can be expressed as:
$A=CU,$
where:

• $C\in {\mathbb{R}}^{m\times r}$ contains $r$ selected columns of $A$.
• $U\in {\mathbb{R}}^{r\times n}$ contains coefficients encoding the linear combination of rows of $C$ to reconstruct $A$.
  Here, $r$ is the rank of $A$ or an approximation to it.

Intuition

CR factorization separates a matrix into:

1. A subset of its columns ($C$), capturing essential structural information.
2. A matrix ($U$) that linearly combines these columns to form the rows of $A$.

This factorization is particularly useful when the matrix $A$ is rank-deficient or approximately low-rank, enabling efficient storage and computations.

Key Properties

1. Rank Preservation: If $A$ has Rank $r$, the product $CU$ also has rank $r$.
2. Efficient Approximation: By selecting $r<\text{rank}(A)$, CR factorization can approximate $A$ with reduced storage.
3. Column-Row Interpretation:
   • $C$ provides representative columns.
   • $U$ encodes how the rows of $A$ can be reconstructed using these columns.
4. Existence: Exists for any matrix $A$; $C$ can be chosen using pivoted column selection or other optimization methods like QR decomposition.

Applications

1. Dimensionality Reduction: Provides a compact representation of data matrices for feature selection and compression.
2. Low-Rank Matrix Approximation: Useful in scenarios like PCA, where approximate low-rank structures are desired.
3. Computational Efficiency: Reduces the cost of operations on large matrices, particularly in numerical linear algebra and machine learning tasks.
4. Data Reconstruction: In applications like collaborative filtering, CR factorization aids in reconstructing missing entries in matrices.
5. Interpretability: The use of actual columns and rows from the data makes the factorization interpretable.