Khatri–Rao product


In mathematics, the Khatri–Rao product is defined as
in which the ij-th block is the sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and column partitions of both matrices is equal. The size of the product is then.
For example, if A and B both are partitioned matrices e.g.:
we obtain:
This is a submatrix of the Tracy–Singh product of the two matrices and also may be called the block Kronecker product.

Column-wise Khatri–Rao product

A column-wise Kronecker product of two matrices may also be called the Khatri–Rao product. This product assumes the partitions of the matrices are their columns. In this case,, and for each j:. The resulting product is a matrix of which each column is the Kronecker product of the corresponding columns of A and B. Using the matrices from the previous examples with the columns partitioned:
so that:
This column-wise version of the Khatri-Rao product is useful in linear algebra approaches to data analytical processing and in optimizing the solution of inverse problems dealing with a diagonal matrix.
In 1996 the Column-wise Khatri–Rao product was proposed to estimate the Angle of arrivals and delays of multipath signals and four coordinates of signals sources at a digital antenna array.

Face-splitting product

The alternative concept of the matrix product, which uses row-wise splitting of matrices with a given quantity of rows, was proposed by V. Slyusar in 1996.
This matrix operation was named the "face-splitting product" of matrices or the "transposed Khatri-Rao product". This type of operation is based on row-by-row Kronecker products of two matrices. Using the matrices from the previous examples with the rows partitioned:
can be get:

Main properties

Examples


Theorem

In particular, if the entries of are can get which matches the Johnson–Lindenstrauss lemma of when is small.

Block Face-splitting Product

According to the definition of V. Slyusar the Block Face-Splitting Product of two partitioned matrices with a given quantity of rows in blocks
can be write as :
The Transposed Block Face-Splitting Product of two partitioned matrices with a given quantity of columns in blocks has a view:

Main properties