Journal of Computational Finance

Risk.net

Adjoint differentiation for generic matrix functions

Andrei Goloubentsev, Dmitri Goloubentsev and Evgeny Lakshtanov

  • Adjoint Differentiation (AD) of matrix operations requiring an SVD/Spectral decomposition (e.g. square root, exp, spectral cut-off).
  • The solution does not require passing adjoints through the SVD/Spectral decomposition.
  • An explicit formula for AD of the Nearest Correlation Matrix routine (NCM).
  • An explicit formula for AD of the Regression Regularization (Tikhonov, spectral cut-off).

We derive a formula for the adjoint Ā of a square-matrix operation of the form f(A), where f is holomorphic in the neighborhood of each eigenvalue.We consider special cases such as the spectral decomposition A = UDU-1 and the spectrum cutoff f(A) = A+ for symmetric A. We then apply the formula to derive closed-form expressions in particular cases of interest to quantitative finance such as the “nearest correlation matrix” routine and regularized linear regression.

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