Low rank tensor computation

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1

I want to generate a rank-1 tensor with Julia for some tensor decomposition. As for SVD and linear combination of singular values and orthogonal basis vectors, one can decompose a matrix by a sum like the sum of singular values * u * vā€™.

I can generate a 10x20x30 matrix like this. (the original code is inspired by GitHub - yunjhongwu/TensorDecompositions.jl: A Julia implementation of tensor decomposition algorithms)

	u = randn(10); v = randn(20); w = randn(30)
	# Generate a noisy rank-1 tensor
	T = cat(map(x -> x * u * v', w)...; dims=3) + 0.2 * randn(10, 20, 30)

I am not sure why I have to use ā€¦ in the formula. I concatenate over the third dimension.

Following code is only giving me a 10x20 matrix:

Su = randn(10); Sv = randn(20); Sw = randn(30)
ST = map(x -> x * Su * Sv', Sw)
2

I tested it with a simpler version. It seems that ā€¦ is converting the array over matrices to an array over arrays:

Su = [1; 2.]; Sv = [2; 3.]; Sw = [3.; 2]
ST = cat(map(x -> x * Su * Sv', Sw)..., dims=3)

with ā€¦

2Ɨ2Ɨ2 Array{Float64, 3}:
[:, :, 1] =
  6.0   9.0
 12.0  18.0

[:, :, 2] =
 4.0   6.0
 8.0  12.0

without ā€¦ producing

2Ɨ1Ɨ1 Array{Matrix{Float64}, 3}:
[:, :, 1] =
 [6.0 9.0; 12.0 18.0]
 [4.0 6.0; 8.0 12.0]
3 
T = [a*b*c for a in u, b in v, c in w]

is also an option. Then adding the randomness could be done with:

T .+= 0.2 .* rand(size(T))
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