Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 KhatriRaoProduct

Source Notebook

Evaluate the Khatri-Rao product of matrices

Contributed by: Jan Mangaldan

ResourceFunction["KhatriRaoProduct"][m1,m2,…]

constructs the Khatri-Rao product of the matrices mi.

Details

The Khatri-Rao product is a columnwise Kronecker product of matrices; that is, each column of the Khatri-Rao product is the Kronecker product of the corresponding columns of the mi.
The matrices mi must all have the same number of columns.

Examples

Basic Examples (1) 

Khatri-Rao product of two matrices:

In[1]:=
am = Array[Subscript[a, ##] &, {2, 2}]; bm = Array[Subscript[b, ##] &, {2, 2}];
In[2]:=
ResourceFunction["KhatriRaoProduct"][am, bm] // MatrixForm
Out[2]=

Scope (2) 

a and b are matrices with exact entries:

In[3]:=
a = {{0, 1}, {-1, 0}}; b = {{1, 2}, {3, 4}};

Use exact arithmetic to compute the Khatri-Rao product:

In[4]:=
ResourceFunction["KhatriRaoProduct"][a, b] // MatrixForm
Out[4]=

Use machine arithmetic:

In[5]:=
ResourceFunction["KhatriRaoProduct"][N[a], N[b]] // MatrixForm
Out[5]=

Use 20-digit precision arithmetic:

In[6]:=
ResourceFunction["KhatriRaoProduct"][N[a, 20], N[b, 20]] // MatrixForm
Out[6]=

Evaluate the Khatri-Rao product of three matrices:

In[7]:=
ResourceFunction["KhatriRaoProduct"][RandomReal[1, {2, 2}], RandomReal[1, {3, 2}], RandomReal[1, {2, 2}]]
Out[7]=

Properties and Relations (3) 

The Khatri-Rao product is multi-linear (linear in each argument):

In[8]:=
m1 = Array[\[FormalX], {4, 3}]; m2 = Array[\[FormalY], {4, 3}]; m3 = Array[\[FormalZ], {4, 3}];
In[9]:=
ResourceFunction["KhatriRaoProduct"][C[1] m1 + C[2] m2, m3] == C[1] ResourceFunction["KhatriRaoProduct"][m1, m3] + C[2] ResourceFunction["KhatriRaoProduct"][m2, m3] // Simplify
Out[9]=
In[10]:=
ResourceFunction["KhatriRaoProduct"][m1, C[1] m2 + C[2] m3] == C[1] ResourceFunction["KhatriRaoProduct"][m1, m2] + C[2] ResourceFunction["KhatriRaoProduct"][m1, m3] // Simplify
Out[10]=

The Khatri-Rao product is associative:

In[11]:=
ResourceFunction["KhatriRaoProduct"][m1, ResourceFunction["KhatriRaoProduct"][m2, m3]] === ResourceFunction["KhatriRaoProduct"][ ResourceFunction["KhatriRaoProduct"][m1, m2], m3] === ResourceFunction["KhatriRaoProduct"][m1, m2, m3]
Out[11]=

The Khatri-Rao product is not commutative:

In[12]:=
ResourceFunction["KhatriRaoProduct"][m1, m2] === ResourceFunction["KhatriRaoProduct"][m2, m1]
Out[12]=

Version History

  • 1.0.0 – 11 April 2022

Source Metadata

  • Citation:
    • Harshman R.A., Hong S., "‘Stretch’ vs ‘slice’ methods for representing three-way structure via matrix notation." Journal of Chemometrics, vol. 16, no. 4, 198-205, 2002.
      DOI: 10.1002/cem.709
    • Khatri C.G., Rao C.R., "Solutions to some functional equations and their applications to characterization of probability distributions." Sankhya, vol. 30, 167-180, 1968.
      Available from       https://www.jstor.org/stable/25049527

Related Resources

Author Notes

The implementation here makes no attempt to exploit sparsity, structure or symmetry of the input matrices.

License Information