Suppose we have a positive semidefinite matrix $H$ with eigenvalues $\lambda_i$ and consider the following ratio of eigenvalue moments:
$$r=\frac{(\lambda_1+\lambda_2+\ldots+\lambda_n)^2}{\lambda_1^2+\lambda_2^2+\ldots+\lambda_n^2}=n \frac{\left(E\lambda\right)^2}{E\lambda^2}$$
This quantity seems to capture the difficulty of minimizing quadratic with Hessian $H$ better than condition number of $H$.
- Does it have a name?
- Does it come up in other interesting applications?
Interpretations I've found so far:
a single step of gradient descent with starting weights initialized from standard normal may reduce expected loss by up $r$ in first step
in the basis of eigenvectors of $X$, transform unit box by $H$, $\sqrt{r}$ now measures the ratio of lengths of longest diagonal using Manhattan vs Euclidian distance (from here)
scale doesn't matter, so can normalize $\lambda$ to add up to 1. If we interpret $H$ as a quantum state with purity $\rho$, $r=1/\rho$. Alternatively, express it in terms of linear entropy of the eigenvalue distribution
$$r=\frac{1}{(1-H_L)}$$
