tensorly.tenalg
.khatri_rao
- khatri_rao(matrices, weights=None, skip_matrix=None, reverse=False, mask=None)[source]
Khatri-Rao product of a list of matrices
This can be seen as a column-wise kronecker product. (see [1] for more details).
If one matrix only is given, that matrix is directly returned.
- Parameters:
- matrices2D-array list
list of matrices with the same number of columns, i.e.:
for i in len(matrices): matrices[i].shape = (n_i, m)
- weights1D-array
array of weights for each rank, of length m, the number of column of the factors (i.e. m == factor[i].shape[1] for any factor)
- skip_matrixNone or int, optional, default is None
if not None, index of a matrix to skip
- reversebool, optional
if True, the order of the matrices is reversed
- Returns:
- khatri_rao_product: matrix of shape
(prod(n_i), m)
where
prod(n_i) = prod([m.shape[0] for m in matrices])
i.e. the product of the number of rows of all the matrices in the product.
- khatri_rao_product: matrix of shape
Notes
Mathematically:
\[\begin{split}\text{If every matrix } U_k \text{ is of size } (I_k \times R),\\ \text{Then } \left(U_1 \bigodot \cdots \bigodot U_n \right) \text{ is of size } (\prod_{k=1}^n I_k \times R)\end{split}\]A more intuitive but slower implementation is:
kr_product = np.zeros((n_rows, n_columns)) for i in range(n_columns): cum_prod = matrices[0][:, i] # Accumulates the khatri-rao product of the i-th columns for matrix in matrices[1:]: cum_prod = np.einsum('i,j->ij', cum_prod, matrix[:, i]).ravel() # the i-th column corresponds to the kronecker product of all the i-th columns of all matrices: kr_product[:, i] = cum_prod return kr_product
References
[1]T.G.Kolda and B.W.Bader, “Tensor Decompositions and Applications”, SIAM REVIEW, vol. 51, n. 3, pp. 455-500, 2009.