QR Factorization #

Check whether Tensor.qr A reconstructs $A$ as $QR$ and gives orthonormal columns in $Q$. The checks use Float with an explicit tolerance. Reduced QR uses Q : Tensor Float [rows, min rows columns] and R : Tensor Float [min rows columns, columns].

Run lake exe torchlean factorizations. The rank-deficient negative control still reconstructs the input, but fails orthonormality: reconstructing a matrix alone does not establish both QR properties. This file tests the implementation; it does not prove a real-arithmetic QR theorem.

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def NN.Examples.Factorization.QR.fullRankMatrix :

TorchLean.Tensor Float [3, 3]

Full-rank matrix used for the positive QR check.

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def NN.Examples.Factorization.QR.fullRankFactors :

TorchLean.Tensor.QRFactors Float 3 3

Compute both factors once.

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def NN.Examples.Factorization.QR.reconstructionError :

Float

Reconstruction error $\lVert A-QR\rVert_{\max}$.

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def NN.Examples.Factorization.QR.orthonormalityError :

Float

Orthonormality error $\lVert Q^\mathsf{T}Q-I\rVert_{\max}$.

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Wide Matrix #

This case also places a dependent column before a later independent column. A reduced implementation that merely truncates the old square factors loses that later basis direction.

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def NN.Examples.Factorization.QR.wideMatrix :

TorchLean.Tensor Float [2, 3]

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def NN.Examples.Factorization.QR.wideFactors :

TorchLean.Tensor.QRFactors Float 2 3

Reduced QR of a wide matrix: Q is 2 x 2 and R is 2 x 3.

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def NN.Examples.Factorization.QR.wideReconstructionError :

Float

How far Q R is from the original matrix; should be at rounding level.

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def NN.Examples.Factorization.QR.wideOrthonormalityError :

Float

How far Qᵀ Q is from the identity, the other half of what QR promises.

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Negative Control #

The orthonormality property requires full column rank. The following matrix has one dependent column. Gram-Schmidt still reconstructs it, but the corresponding column of Q vanishes and $Q^\mathsf{T}Q\ne I$.

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def NN.Examples.Factorization.QR.rankDeficientMatrix :

TorchLean.Tensor Float [3, 3]

A matrix whose second column is twice its first.

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def NN.Examples.Factorization.QR.rankDeficientFactors :

TorchLean.Tensor.QRFactors Float 3 3

QR of a rank-deficient matrix. Reconstruction survives; orthonormality does not, because the dependent column contributes a zero column to Q.

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def NN.Examples.Factorization.QR.rankDeficientReconstructionError :