A service stores each user as a two-number record and wants to compress every record down to a single number using PCA. The pipeline is fitted once, on these five training records:
| User | ||
|---|---|---|
| A | ||
| B | ||
| C | ||
| D | ||
| E |
Fitting means: centre the training records, build their covariance matrix, and take the unit eigenvector of its largest eigenvalue as the PC1 direction.
A new user record then arrives at . It is not used to fit anything — the mean and the covariance matrix stay exactly as the five training records defined them. The new record is pushed through the finished pipeline:
How far is the decompressed point from the original record ?
Give the Euclidean distance, rounded to two decimal places. Carry at least four decimal places through the intermediate steps and round only at the end.
Toggle to project every point onto the principal component.
Instead of storing two coordinates, we now store one position along the line. The important pattern remains.
Once you get it right, the reasoning behind the answer shows up here.