I am new here, and a physicist, so excuse me in case I do not use the right jargon but I think this is a question for the math community:
In 1-D, I have a signal, represented by a vector v of dimension n, that is "deformed" by a matrix A (m,n) to give w (m): $$ \mathbf{w} = A\mathbf{v} $$
In my experiment I measure w and I know A, so to estimate v I do $$ \mathbf{v'} = A^+\mathbf{w} $$ $A^+$ being the (Moore-Penrose)pseudo-inverse. A has independent rows, $m<n$, $A^+$ is right-inverse (these statements are equivalent, right?).
When I simulate the experiment v' gives a reasonable estimate of v. I interpret $\mathbf{v'}=A^+A\mathbf{v}$ as losing information in $A$ and a kind of interpolation in $A^+$.
Now I would like to do the same thing, but where my signal is a 2-D image, and the deformation is described by a 4-th order tensor. So the image "v" now has dimensions (n,n), the measurement w (m,m) and thus A has now become a tensor of dimensions (m,m,n,n).
Is there such a thing as a pseudoinverse for a 4-th order tensor? If not, why not?
(If yes, a link to a (python-)code would be much appreciated)