represent hankel matrix by low rank tensorial approximation

Question

suppose that we have given following matrix

\begin{matrix} x_1 & x_2 & ..x_p \\ x_2 & x_3 & ...x_{p+1} \\ . & .& . & \\ x_{N-p+1} & x_{n-p+2} &... x_n \end{matrix}

where $N$ is length of $x$ vector,namely $x={x_1,x_2,..x_N}$

and $p$ is parameter chosen such that $1<p<N$,we can represent given matrix by following tonsorial approximation

$\sum\limits_{i=1}^{min(p,l)} \lambda_i*(U_i \bigotimes V_i ) $

this represent tensorial approximation of original matrix,where $U_i$ and $V_i$ are singular vectors from this matrix and $\lambda_i$ is singular value,in this case we are taking columns of singular matrices,my question is following how can i express each $x_i$,that is $x={x_1,x_2,...x_N}$ from this formula?in my mind $x_1$ will be first element of matrix got by sum of all this matrix,this will be for $x_p$ as well,but what about $x_{p+1}$ ?can you help me please to finish this formula?

UPDATED : for instance

>> a=[2 1 3;1 2 3];
>> a

a =

     2     1     3
     1     2     3

>> [U E V]=svd(a);
>> x1=kron(U(:,1),V(:,1));
>> x2=kron(U(:,2),V(:,2));
E(1,1)*x1+E(2,2)*x2

ans =

    2.0000
    1.0000
    3.0000
    1.0000
    2.0000
    3.0000

Answer 1

If $A=USV^T$ is an SVD of a matrix $A\in\mathbb{R}^{m\times n}$, then $$\tag{1} a_{ij}=e_i^TAe_j=e_i^TUSV^Te_j=\sum_{k=1}^r(e_i^TUe_k)(e_k^TSe_k)(e_k^TV^Te_j)=\sum_{k=1}^r s_k u_{ik}v_{jk}, $$ where $s_k$ is the $k$th diagonal element of $S$ and $r$ is the rank of $A$.

This is true for the "exact" SVD, not necessarily the truncated one as of course the truncated SVD approximation of $A$ is not equal to $A$. It also does not need to have the Hankel structure unless $A$ is square (and hence symmetric).