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I am wondering what is the second order Taylor expansion of a vector-valued function $f(x):\mathbb{R}^M\rightarrow \mathbb{R}^N$.

I know that the gradient of a vector-valued function is a Jacobian matrix $\nabla f(x)\in \mathbb{R}^{M\times N}$, and using this we have the first order Taylor expansion $f(x) = f(y)+\nabla f(y)^\top (x-y) + \text{higher order terms}$.

My question is: what is the second-order expansion? It seems that the "Hessian" of $f(x)$ is a $M\times N \times M$ matrix (tensor). Do I need some tensor operations to form the expansion?

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  • $\begingroup$ You can either use tensor products (see Wikipedia's page on jets) or just look at the expression term by term using multi-index notation (see Wikipedia's page on Taylor's theorem). $\endgroup$ – user137731 Nov 17 '15 at 1:02
  • $\begingroup$ Thanks for your comments. Tensor product approach seems interesting. I will look into it. $\endgroup$ – user3138073 Nov 18 '15 at 2:25

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