The discussion revolves around finding the gradient and Hessian of a function defined by two n-dimensional vectors, f(𝑥,𝑦). Participants explore the mathematical treatment of this function in terms of its derivatives.
Participants generally agree on the approach to deriving the gradient and Hessian, but the discussion includes varying levels of detail and interpretation regarding the mathematical structures involved.
Some assumptions about the definitions of the gradient and Hessian in this context may not be explicitly stated, and the discussion does not resolve all nuances regarding the treatment of the function's derivatives.
mathwonk said:i would just treat it as a function of 2n variables and take the vector of 1st, then the matrix of 2nd derivatives.
mathwonk said:yep. basically the gradient is a linear map approximating the original map. so you could also view it as a linear map from R^n x R^n -->R, and the Hessian I suppose as a bilinear such map.
I.e. if R^n = V, the 1st derivative is a linear map VxV-->R, and the second derivative is a symmetric bilinear map (VxV)x(VxV)-->R.
So if you really want to break up your source space into a pair of n vectors, then
you get two "partial" gradients, each a 1byn matrix, and you get a 2 by 2 Hessian matrix, where each block is an nbyn matrix, i.e. 4 nbyn matrices of vector second derivatives