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 week-1
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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.