I came across such concept: If integral over second derivative of function $g : R \rightarrow R$ : $\int_{-\infty}^{\infty} [g''(x)]^2dx$ is high, then function is more "wiggly", and the lower value of this integral the more smooth is function. How to apply this concept to multivariable function $f : R^{n} \rightarrow R$? Calculate the integral over matrix norm of hessian matrix of this function?
Could you suggest either heuristical or strictly theoretical approach to solve this problem?
Actually, integrating $g''(x)$ doesn't work well as a measure since you can have cancellation if there are places where $g$ is convex and other places where $g$ is concave. Rather, you'd typically regularize by minimizing
$\int_{-\infty}^{\infty} g''(x)^{2} dx$
In higher dimensions, the Laplacian of $g$ is typically used:
$\int_{R^{n}} (\Delta g(x))^{2} dx $
Here
$\Delta g(x)= \frac{\partial^{2} g(x)}{\partial x_{1}^{2}} + \cdots + \frac{\partial^{2} g(x)}{\partial x_{n}^{2}}$.