The hat notation signifies that the reconstructed point is an approximation lying in the span of the retained principal directions, not necessarily the exact original point. Compression inevitably discards information, so the reverse mapping cannot perfectly recover the original measurements.
Conditions: A higher-dimensional dataset is mapped to a lower-dimensional representation.; Reconstruction is attempted from the lower-dimensional coordinates.
The hat notation signifies that the reconstructed point is an approximation lying in the span of the retained principal directions, not necessarily the exact original point. Compression inevitably discards information, so the reverse mapping cannot perfectly recover the original measurements.
Conditions: A higher-dimensional dataset is mapped to a lower-dimensional representation.; Reconstruction is attempted from the lower-dimensional coordinates.