Calculation with three dimensional matrices

I want to calculate a vector , with each element defined as follows

, where is the set of N dimensional vector, and is also the set of N dimensional vector, c is a constant. The superscript T is denoted as the transpose operation.

In my matlab code, is stored as a three dimensional matrix G, its dimension is (where ).

is also stored as a three dimensional matrix W, its dimension is (where ). c is a scalar.

I know i can use multiple for loop to calculate this vector Y, but it is too inefficient. Is there any some fast way to calculate it, maybe use three-dimensional matrix calculation method?

[Matt J]

Isn't supposed to represent a scalar product? If so, why do you have a subscript of 2 in as if you are taking l2-norm of a vector?

[Hancheng Zhu]

oh，i make a mistake, it is just a absolute value of this complex scalar, let me revise it.

[埃博拉酱]

g是矩阵，不是个标量吗？你对标量做转置是什么意思？我感觉你写的公式还是有问题，不如直接上代码……

g is a matrix, not a scalar? What do you mean by transposing a scalar? I feel that there is still something wrong with the formula you wrote. It is better to go directly to the code...

[Torsten]

@埃博拉酱

is an N-dimensional vector.

[埃博拉酱]

@Torsten Then he is transposing a 1×1×N vector (since G is K×J×N). I don't think transposition is well-defined for a vector on dimension 3.

[Torsten]

It's just a formal mathematical description of what should be coded. In the code, g_(k,j) and w_(k,i) must be of size (N,1).

[埃博拉酱]

@Torsten I've always felt that mathematical formulas are far less expressive than program code. Many of the symbolic representations are rudimentary, crude, and ambiguous. Shortness is perhaps its only "advantage" - which is often criticized in program code for sacrificing readability.

[Torsten]

I've made the opposite experience.

Imagine you have code for the computation of Y_i. I would be really surprised if it were easier to read than the mathematical formula from above.

[Paul]

@Hancheng Zhu

Perhaps you should post the code that you have along with sample input data.

Answer 1

[埃博拉酱]

If your formula expresses what @Torsten understands, I'll code like this:

function Y=cGW_Y(c,G,W)
Y=permute(pagemtimes(permute(G,[2,3,1]),permute(W,[3,2,1])),[4,3,2,1]);%1×K×I×J
Y=pagemtimes(Y,'none',Y,'ctranspose');%1×1×I×J
Y=log2(sum(Y./(sum(Y,3)+c*c-Y),4)+1);%1×1×I
end

[Hancheng Zhu]

yes, this is correct.

Answer 2

[Divyanshu]

Hello @Hancheng Zhu,

You can try using 'pagetimes' function of MATLAB. For more details about 'pagetimes' refer the following documentation link:

https://www.mathworks.com/help/matlab/ref/pagemtimes.html

Additionally, you can take reference from following MATLAB answer thread as well:

https://www.mathworks.com/matlabcentral/answers/755404-3d-matrix-multiplication

Hope it helps!