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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 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 $\mathbb{R}^3$.