Why is the reconstructed point denoted with a hat as $\hat{\mathbf{x}}$ instead of the original $\mathbf{x}$?

Conditions

• A higher-dimensional dataset is mapped to a lower-dimensional representation.
• Reconstruction is attempted from the lower-dimensional coordinates.

Reasoning, step by step

1. Acknowledge that dimensionality reduction by projection loses information.
2. Perform the reconstruction map from 2D coordinates back to 3D space.
3. Observe that the reconstructed point lies strictly on the 2D subspace.
4. Conclude that unless the original point was already on the subspace, the reconstruction is an approximation.
5. Use the hat notation $\hat{\mathbf{x}}$ to denote this approximate recovery.

Example

The video explicitly uses $\hat{\mathbf{x}}$ in the reconstruction formula and the narrator states that compression inevitably discards information, asking if the journey can be reversed to rebuild original measurements.

Common misconceptions

• Assuming that PCA reconstruction is a lossless process that perfectly restores the original data.
• Believing that the reconstructed points will magically recover their original heights or depths outside the subspace.

Watch the explanation

Explore next

• How is the reconstruction error defined and visualized in the video?
• What does the direction of the reconstruction error correspond to in PCA?