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ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A rank-one centered data matrix is reconstructed exactly by rank-one PCA

Example

Let

X=(1000−10)∈R3×2,q=1.

The columns of X already have mean zero and X has rank one. The rank-one PCA frame Q=(1,0)T reconstructs X exactly, so the minimum squared Frobenius reconstruction error is zero and the unique minimizing subspace is span⁡{(1,0)T}.

Facts & Assumptions

Given: The stated 3×2 data matrix X and target rank q=1.

[F1]

Rank-q PCA centers the data, minimizes ∥Xc−XcQQT∥F2 over QTQ=Iq, and reconstructs by XcQQT (Rank-q principal component analysis by Frobenius reconstruction error).

[F2]

The global minimum error is the sum of squared singular values after index q, and minimizing subspaces are the top-q right singular subspaces (The top right singular subspace minimizes rank-q PCA Frobenius reconstruction error).

Verification

1.1givenF1algebra

Each column of X sums to zero, so Xc=X; multiplication gives XTX=(2000), with singular values 2 and 0.

2.1step 1.1F1algebra

The vector Q=(1,0)T satisfies QTQ=1, and XcQQT=X. Therefore its reconstruction error is zero.

3.1step 2.1F2algebra∎

Every squared Frobenius error is nonnegative, so this zero error is the global minimum. Since 2>0 separates the two eigenvalues of XTX, [F2] also gives the unique minimizing one-dimensional subspace span⁡{(1,0)T}.

Depends on

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