Let's say I have an image representing a sampled function. It just so happens that I know this function can be represented as a sum of individual outer products along with some noise. So I might have an image defined as:
$I(x,y) = a_0b_0^T + a_1b_1^T + ... +\ \Omega$
Where the a and b vectors are the individual vectors forming the outer products, and $\Omega$ is a matrix of IID gaussian random variables.
If I decompose $I(x,y)$ using the SVD, I similarly get a sum of outer products:
$I(x,y) = U \Sigma V^T = s_0u_0v_0^T + s_1u_1v_1^T + ... s_Nu_Nv_N^t$
I would like to be able to extract the original image components (neglecting the noise), given these "singular images". Is there anything that I can confidently say about the relationship between my source inner products and output inner products?