[removed]

https://en.wikipedia.org/wiki/Elastic_net_regularization

If your goal is to do linear regression and enforce hard sparsity constraints on W, then there are several algorithms to do this directly (not guaranteed to recover the true sparse W unless certain conditions are met though). A simple starting point might be orthogonal matching pursuit: https://scikit-learn.org/stable/auto_examples/linear_model/plot_omp.html.

Are you asking about the feasibility ridge regression with sparse inputs? Or about regularization to enforce a sparse solution?

Basically is the feasibility ridge regression with sparse inputs, but I want to select partial units of W acting on A and B. For instance, if I have A of (2x5) and B of (5x5) and I choose units 2 and 4, the columns [0,1,3] of A are zeros and columns and rows of B with index [0,1,3] are also zero. I select the units 2 and 4 with some importance mechanism. The question is: there is a way of having W* resulting from filter A and B that is similar to W computed without filtering A and B?

I asked because filtering A and B break the inversion and so the computation of W. I don't know if there exists some way of decomposing B in order to invert more easily or something like this.

Anyway thanks for your interest!

You can still compute the estimator with sparse inputs because the regularization term ensures the denominator is full rank. If the zeros are standing in for missing values, however, your estimates will be biased.

As for your second question, W* computed from only columns 2 and 4 will only yield the same values as W in the unrestricted model if the columns of X are orthogonal. Could you work with an orthogonal transform (e.g. PCA projection) of the X matrix?