Use of moore-penrose inverse when modeling PCA. In the derivation for principal component analysis we model our observed data points y as the result of a linear transformation restricted to being an axis change, W, applied to a set of uncorrelated variables x, where x lives in a lower dimensional space than y.

Amy Bright

Amy Bright

Answered question

2022-11-14

Use of moore-penrose inverse when modeling PCA
In the derivation for principal component analysis we model our observed data points y as the result of a linear transformation restricted to being an axis change, W, applied to a set of uncorrelated variables x, where x lives in a lower dimensional space than y.
Thus, we can represent our observations as y = W x, and what we are trying to find as x = W y, where W' is the pseudo-inverse of W.
Is the reason we choose to model the mapping of y onto the pca axes with use of a pseudo-inverse due to the fact that we want to be able to use PCA on systems where y doesn't completely follow our assumptions? Where not all points in y are contained in the column space of W?

Answer & Explanation

Envetenib8ne

Envetenib8ne

Beginner2022-11-15Added 17 answers

Step 1
If you're familiar with linear regression, there's not much new going on here.
Step 2
The quantity x = W y is the solution to W x = y which minimizes y W x 2 2 . Minimizing that quantity means finding the subspace with dimension equal to the number of columns of W such that the data have the lowest squared L 2 distance from it.

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