How is the reconstruction error for a single data point defined in the video?

Conditions

• The original data point $\mathbf{x}_i$ is in 3D space.
• The reconstructed point $\hat{\mathbf{x}}_i$ is on the 2D subspace.
• Distance is measured using the Euclidean norm.

Reasoning, step by step

1. Identify the original 3D data point $\mathbf{x}_i$.
2. Identify the reconstructed 3D point $\hat{\mathbf{x}}_i$ obtained from 2D coordinates.
3. Calculate the difference vector between the original and reconstructed points.
4. Apply the Euclidean norm to this difference vector.
5. The resulting scalar is the reconstruction error $e_i$.

Example

The video displays the formula $e_i = \|\mathbf{x}_i - \hat{\mathbf{x}}_i\|$ and visualizes the error as red vertical whiskers connecting floating cyan points to the blue projection plane.

Common misconceptions

• Assuming the error is measured along the axes of the original coordinate system rather than the shortest path.
• Believing the error is zero for all points if the subspace captures most of the variance.

Watch the explanation

Explore next

• In what direction does the reconstruction error point in PCA?
• How is the total reconstruction error across all points calculated?