Sensitivity of predicted value to features

In linear regression, how to determine and order the features in terms of the magnitude of their contribution to the predicted value. Order features by significance
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This is not really necessary. Do you have some reason for doing this?

One clue as to the importance of each feature is to look at the magnitude and sign of its weight value after training to minimize the cost.

I am trying to determine the major features and potentially reduce the number features based on that insight . My math is not fresh but I think in SVD, you can order the eigenvalues and reduce matrix dimension based on it? What I am asking for is analogous to this except perhaps can hopefully avoid inversion of a large matrix

Yes, you can use PCA (which is based on SVD) to reduce the dimensionality.
It requires inverting a potentially very large matrix of all of your training examples.

Whether this is feasible depends on the size of the data set and the capability of your computing platform.

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