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pythonnumpyscipysvd
shaifali Gupta
shaifali Gupta

Reputation: 388

Python: get unsorted list of singular values from numpy/scipy svd

I have a square matrix and want to use svd to reduce condition number of the matrix by elimination of some rows/columns.

I used numpy/scipy both give sorted list of singular values.

Using sorted list, I can easily reconstruct a smaller matrix by discarding some small singular values. But it is difficult to map to the original matrix, that what values have been eliminated and what values have been retained. I need that further.

Is there any way to identify later for the original matrix, what indices have been retained and what discarded.

Upvotes: 1

Views: 1699

Answers (1)

FChm
FChm

Reputation: 2590

To perform a singular value decomposition of a matrix you can look at the .linalg` module in numpy.

A SVD of a matrix factorizes it into the product of three matrices:

M = U S V*

M is your original matrix. S is a rectangular diagonal matrix with the ('sorted') singular values on the diagonals. U and V are known as the left- and right-singular vectors respectively.

Note: np.linalg.svd doesn't return S but s which is just a 1D array containing the singular values.

Practical implementation

Lets say you have an (m x q) feature space represented by the 2D array X, where X is a centered matrix. You can calculate its SVD:

U, s, Vt = np.linalg.svd(X)

where the t denotes the transpose of V and s is your 'unsorted list of singular values'.

You can then project your original feature space to n dimensions by using the singular vectors and discarding singular vectors which preserve the least variance: 

X_projected = X.dot(Vt.T[:,:n])

where X_projected is now the representation of your feature space in the lower n-dimensional space.

Importantly, you can transform back from your reduced feature space to your original space:

X_recovered = X_projected.dot(Vt[:,:n])

Notably, this can be used to measure the information lost in your reduced feature set by comparing X_recovered to your original feature set (X) to measure things such as reconstruction error.

Upvotes: 1

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