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
The gradient of a function of two vectors is a vector that represents the rate of change or slope of the function at a particular point. It is a vector formed by taking the partial derivatives of the function with respect to each of the input vectors.
The gradient is calculated by taking the partial derivative of the function with respect to each input vector and combining them into a vector. This can be represented mathematically as ∇f(x,y) = [∂f/∂x, ∂f/∂y].
The Hessian of a function of two vectors is a matrix that contains second-order partial derivatives of the function. It provides information about the curvature of the function at a specific point and is used to determine if the point is a local minimum, maximum, or saddle point.
The Hessian is calculated by taking the second-order partial derivatives of the function with respect to each input vector and arranging them into a matrix. This can be represented mathematically as H = [∂²f/∂x², ∂²f/∂x∂y; ∂²f/∂y∂x, ∂²f/∂y²].
The gradient and Hessian are important in optimization because they provide information about the direction and rate of change of a function. This information can be used to find the minimum or maximum of a function, which is often the goal in optimization problems.