Tensor example

Define a tensor based on an irrational function, disturbed by a random number

Start by defining your model H $$ H(x_1,x_2) = \frac{5x_2^2}{2+\sin(4x_1)+x_2^4} (1+r), $$ where $r\in[-0.075,+0.075]$ is a random number.

%%% Tensor model generation
H   = @(x) 5*x(:,2).^2./(2+sin(4*x(:,1))+x(:,2).^4);
n   = 2;
ip  = {linspace(-pi,1,60) linspace(-1,2,50)};
%%% Column/Row
for i = 1:n
	p_c{i} = ip{i}(2:2:end);
	p_r{i} = ip{i}(1:2:end);
end
%%% Data tensor/rand
[y,x,dim]   = mlf.make_tab_vec(H,p_c,p_r);
y           = y.*(1+.075*(rand(length(y),1)-.5));
tab         = mlf.vec2mat(y,dim);

Then, the tensorized data tab are constructed. Here leading to a 2-D tensor, and more specifically $$ \mathcal T_2^{\otimes} \in \mathbb R^{60\times 50} $$

Use the mLF (Alg. 1: direct [A/G/P-V, 2025])

Now we are ready to build the approximation. The opt.method is used to select either the recursive ('rec') or full ('full') method. Here the latter is used and results in better results (still to be understood). The opt.data_min is used to use all data in the null-scapce computation (requires more CPU but may result in a least square averaging).

opt.method_null = 'svd0'; % null space method
opt.method      = 'full'; % full or recursive method
opt.ord_show    = true;   % show order detection step
opt.ord_obj     = [6 4];  % set objective orders
opt.data_min    = false;  % use all the data

Notice that, due to noise, the singular values decay along the both variables is very slow and truncation is hard to detects. However, regarding the first one $x_1$, a decay around 6 is visible. Regarding $x_2$, same remark for 4 holds. Playing with opt.ord_obj allows to adjust this.

Evaluate the resulting approximate

Now we compare the original data and approximate functions. The below figure compares the original data tab and approximate g functions along $x_1$ and $x_2$. The frames show the value (left) and the mismatch (right).