Suppose I'm able to obtain values of $f(0),f(1),f(2),\ldots,f(m)$ with $f$ defined as follows: $$f(s)=\sum_{i=1}^n \exp(-h_i s)h_i$$
Is there a numerical procedure to obtain $h_1,\ldots,h_n$? We know that $h_{i}>h_{i+1}$ and that $h_i>0$. In practice $n>10^6$ and $f(0)$ can be bounded independently of $n$ (hence $h_i$ decay to 0), so $\int_1^\infty$ is a good approximation of $\sum_{i=1}^n$.
background: $f(0),f(1),\ldots$ are multiples of loss observed while doing gradient descent on quadratic with eigenvalues $h_i$. Solving this problem would let you get Hessian spectrum from sequence of loss values. Some extra regularity assumption may need to be added to make this useful. (mathematica.SE crosspost with numeric examples).