Suppose we are given two column vectors, one containing values and the other containing indices referencing to columns of a matrix. I'd like to somehow prune the matrix using the column vectors and eliminate some of its columns which are data samples. I can do it for a specific data by ploting and see which data samples appear to be less "useful", but a generalized rule is what I cannot come up with. The difficulty rise from the fact that one is not aware of the values that stored in one of those column matrix and can't really get a meaningful description of them. I thought mean was helpful but then I don't know what is the relation between mean and values. I am looking for ideas as to which columns (indices in one of those column vectors) must be eliminated.
To better explain how those column vectors were achieved, in the following optimization problem with a sparsity constraint on $\alpha$, matrix $\alpha$ was computed then row-rise $\mathcal{l}_1$ norm was calculated to achieve a column vector which is sorted afterwards to have the above mentioned column vectors:
\begin{equation} \min_{\alpha} \lVert y_i\alpha-m_i\rVert \quad \mathrm{subject \:to} \quad \lVert \alpha\rVert_0 \leq T \end{equation}
$y_i$ is a column of dataset, a data sample.