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NN.Examples.Factorization.Cholesky

Cholesky Factorization #

Tensor.cholesky A returns a lower-triangular factor candidate. Here we factor a 3×3 symmetric positive-definite matrix and check that its Float reconstruction error is small.

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def NN.Examples.Factorization.Cholesky.positiveDefiniteMatrix :
TorchLean.Tensor Float [3, 3]

Symmetric positive-definite matrix used for the positive Cholesky check.

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    def NN.Examples.Factorization.Cholesky.lowerFactor :
    TorchLean.Tensor Float [3, 3]

    Lower-triangular Cholesky factor.

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      def NN.Examples.Factorization.Cholesky.reconstructionError :
      Float

      Reconstruction error $\lVert A-LL^\mathsf{T}\rVert_{\max}$.

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        Negative Control #

        Cholesky requires positive pivots. The matrix below is symmetric but not positive-definite (eigenvalues 3 and -1), so the Float computation reaches the square root of a negative value and the reconstruction error becomes NaN.

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        def NN.Examples.Factorization.Cholesky.indefiniteMatrix :
        TorchLean.Tensor Float [2, 2]

        A symmetric but indefinite matrix (eigenvalues {3, -1}), outside Cholesky's domain.

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          def NN.Examples.Factorization.Cholesky.indefiniteFactor :
          TorchLean.Tensor Float [2, 2]

          The negative pivot produces a NaN factor entry, making reconstruction fail.

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            def NN.Examples.Factorization.Cholesky.indefiniteReconstructionError :
            Float

            Reconstruction error for the indefinite case, which should come out NaN.

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              def NN.Examples.Factorization.Cholesky.check :
              IO Unit

              Run the positive reconstruction check and its indefinite-matrix negative control.

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