Considering regular matrix approximation inequality
|| $A - QQ^TA $|| < e
where we try to approximate matrix $A$ by a lower rank orthonormal matrix $Q$. I've read an article on probabilistic algorithms for such an purpose. You simply pick random (Gaussian etc.) vectors $\omega$, then generate vectors from the range of $A$ by $y =A\omega$. Finally, construct $Q$ by these vectors, $y$, after Gram-Schmidt etc. Details can be found in "Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions".
My question is the following:
Let's assume we have several matrices $A_i$ with "same structure" and each have the same rank $k$. I may give details on the structure if you interest. And we find $Q_i$ for each corresponding matrix $A_i$. Again assume we may define a distribution over $Q_i$ depending on the distribution we use during approximating matrices $Q_i$ (the distribution of $\omega$).
Then, we get a new matrix $A$ with again same structure and rank $k$. But this time we only have a submatrix $A(:,J)$ i.e. only several columns. I want to approximate $Q$ for this new matrix using only given columns.
To summarize we have two phase:
1. We somehow learn a model for $Q$ using given full matrices $A_i$
2. Given a subsample $A(:,J)$ of a novel matrix $A$ (that is not present in the first phase), we derive its corresponding $Q$
I hope, you may simply recommend me some useful references. I do not even know where to look.
Thank you for your time.