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I am trying to get the derivative of this function for use in a numerical optimization problem:

$$f(\mathbf{T}) = ||\mathbf{TY_+}||_* + ||\mathbf{TY_-}||_* - ||\mathbf{T[Y_+, Y_-]}||_*$$

where $\mathbf{T, Y_+, Y_-}$ are matrices, $||\dot{} ||_*$ is the nuclear norm (sum of singular values), and $[\cdot, \cdot]$ is concatenation of matrices.

I attempted to use WolframAlpha but as far as I can google it seems to not be able to do it. I don't have any experience with matrix calculus and it all seems a bit over my head for me to attempt to solve. Can anyone point me to some tool that would be able to derive this for me? Alternatively if anyone cleverer than me wants to take a crack at it they are welcome to.

Rodrigo de Azevedo
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timleathart
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1 Answers1

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Given the differential of the nuclear norm as $$ d\,\|X\|_{*} = X(X^TX)^{-1/2} : dX $$ Simply substitute $X=TY_{+}$ to find the derivative of the first term as $$ \eqalign { d\,\|TY_{+}\|_{*} &= TY_{+}(Y_{+}^TT^TTY_{+})^{-1/2} : dT\,\,Y_{+} \cr &= TY_{+}(Y_{+}^TT^TTY_{+})^{-1/2}Y^T_{+} : dT \cr \cr \frac {\partial \,\|TY_{+}\|_{*}} {\partial T} &= TY_{+}(Y_{+}^TT^TTY_{+})^{-1/2}Y^T_{+} \cr }$$ Now do the same thing for the other 2 terms to arrive at $$ \eqalign { \frac {\partial f} {\partial T} &= TY_{+}(Y_{+}^TT^TTY_{+})^{-1/2}\,Y_{+}^T\cr &+ \, TY_{-}(Y_{-}^TT^TTY_{-})^{-1/2}\,Y_{-}^T\cr &- \, T[Y_{+},Y_{-}]\bigg([Y_{+},Y_{-}]^TT^TT\,[Y_{+},Y_{-}]\bigg)^{-1/2}[Y_{+},Y_{-}]^T\cr }$$

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greg
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