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The reconstruction error for a single data point is defined as the Euclidean distance between the original 3D point and its reconstructed approximation on the 2D subspace.

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.

The reconstruction error for a single data point is defined as the Euclidean distance between the original 3D point and its reconstructed approximation on the 2D subspace.

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.

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.

The reconstruction is formulated as a linear combination of the principal component direction vectors, weighted by the stored 2D coordinates. This is equivalent to multiplying the 2D coordinate vector by a reconstruction matrix whose columns are the principal components.

Conditions: A compressed coordinate vector $\mathbf{z} \in \mathbb{R}^2$ is available.; Reconstruction uses principal component directions $\mathbf{v}_1, \mathbf{v}_2$.; The output is an approximation $\hat{\mathbf{x}}$ in $\mathbb{R}^3$.

The reconstruction is formulated as a linear combination of the principal component direction vectors, weighted by the stored 2D coordinates. This is equivalent to multiplying the 2D coordinate vector by a reconstruction matrix whose columns are the principal components.

Conditions: A compressed coordinate vector $\mathbf{z} \in \mathbb{R}^2$ is available.; Reconstruction uses principal component directions $\mathbf{v}_1, \mathbf{v}_2$.; The output is an approximation $\hat{\mathbf{x}}$ in $\mathbb{R}^3$.