pseudo inverse of sparse matrix in python

Question

I am working with data from neuroimaging and because of the large amount of data, I would like to use sparse matrices for my code (scipy.sparse.lil_matrix or csr_matrix).

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Answer 1

You could study more on the alternatives offered in scipy.sparse.linalg.

Anyway, please note that a pseudo-inverse of a sparse matrix is most likely to be a (very) dense one, so it's not really a fruitful avenue (in general) to follow, when solving sparse linear systems.

You may like to describe a slight more detailed manner your particular problem (dot(A, x)= b+ e). At least specify:

• 'typical' size of A
• 'typical' percentage of nonzero entries in A
• least-squares implies that norm(e) is minimized, but please indicate whether your main interest is on x_hat or on b_hat, where e= b- b_hat and b_hat= dot(A, x_hat)

Update: If you have some idea of the rank of A (and its much smaller than number of columns), you could try total least squares method. Here is a simple implementation, where k is the number of first singular values and vectors to use (i.e. 'effective' rank).

from scipy.sparse import hstack
from scipy.sparse.linalg import svds

def tls(A, b, k= 6):
    """A tls solution of Ax= b, for sparse A."""
    u, s, v= svds(hstack([A, b]), k)
    return v[-1, :-1]/ -v[-1, -1]

Answer 2

Regardless of the answer to my comment, I would think you could accomplish this fairly easily using the Moore-Penrose SVD representation. Find the SVD with scipy.sparse.linalg.svds, replace Sigma by its pseudoinverse, and then multiply V*Sigma_pi*U' to find the pseudoinverse of your original matrix.