MFPCA (version 1.3-10)

ttv: Tensor times vector calculation

Description

Functionality adapted from the MATLAB tensor toolbox (https://www.tensortoolbox.org/).

Usage

ttv(A, v, dim)

Value

An array, the result of the multiplication.

Arguments

A

An array.

v

A list of the same length as dim.

dim

A vector specifying the dimensions for the multiplication.

Details

Let A be a tensor with dimensions \(d_1 \times d_2 \times \ldots \times d_p\) and let v be a vector of length \(d_i\). Then the tensor-vector-product along the \(i\)-th dimension is defined as $$B_{j_1 \ldots j_{i-1}j_{i+1} \ldots j_d} = \sum_{i=1}^{d_i} A_{j_1 \ldots j_{i-1} i j_{i+1} \ldots j_d} \cdot v_i.$$ It can hence be seen as a generalization of the matrix-vector product.

The tensor-vector-product along several dimensions between a tensor A and multiple vectors v_1,...,v_k (\(k \le p\)) is defined as a series of consecutive tensor-vector-product along the different dimensions. For consistency, the multiplications are calculated from the dimension of the highest order to the lowest.

References

B. W. Bader and T. G. Kolda. Algorithm 862: MATLAB tensor classes for fast algorithm prototyping, ACM Transactions on Mathematical Software 32(4):635-653, December 2006.

See Also

UMPCA

Examples

# create a three-mode tensor
a1 <- seq(0,1, length.out = 10)
a2 <- seq(-1,1, length.out = 20)
a3 <- seq(-pi, pi, length.out = 15)
A <-a1 %o% a2 %o% a3
dim(A)

# multiply along different dimensions
dim(ttv(A = A, v = list(rnorm(10)), dim = 1))
dim(ttv(A = A, v = list(rnorm(20)), dim = 2))
dim(ttv(A = A, v = list(rnorm(15)), dim = 3))

# multiply along more than one dimension
length(ttv(A = A, v = list(rnorm(10), rnorm(15)), dim = c(1,3)))