Rectangular matrix non oriented inverse approximation

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Rectangular matrix non oriented inverse approximation

Postby antcout » Tue May 24, 2016 10:23 am

Hello,

I am currently trying to use linear algebra to solve equations involving probability distributions and marginalization : P(X | Y) = \sum_z P(X | z) P(z | Y).
Such marginalization can be done using matrices.
Let denote A the matrix encoding P(X | Y), B the matrix encoding P(X | Z) and C for P(Z | Y).
I now have 3 columns stochastic matrices (summing at 1 for each column) and the above marginalization is equivalent to A = B.C in linear algebra.

My objective is to find P(Z | Y) using this equality, thus try to compute C by the equality B^-1 A = B^-1 B C = C.
However, since the B matrix is not squared, due to discrete random variables with different domain sizes, the inverse does not exist.
Thus, i tried to substitute it with the Moore Penrose pseudo-inverse : B^+ A = B^+ B C.
However again, it appears that i have less values in my random variables associated to rows than the one associated to columns.
As a consequence, the pseudo inverse i compute is automatically a right one. Thus, i can have guarantees on the product B B^+ but not B^+ B which is of interest to me.
Of course, if i change the domain of the problematic random variable, to have a B squared matrix or full rank by column, i can compute what i want.
But it seems "unfair" that the solution of my problem be dependent on the representation i have of my random variables !!

My question is thus the following : is it possible to obtain a left pseudo inverse instead of a right one with a rectangle full rank matrix by rows ?
Or, can we compute an approximation of the inverse which is not oriented, so that i can use it on the left hand size of my equation?
I know this is quite contrary to what Linear Algebra says in books but maybe there is something we can still do in practice.

Thanks in advance for your help.

AC.
antcout
 
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