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Under some reasonable assumptions integrals with large exponents can often be computed via saddle point approximations, e.g. $$\int e^{-\lambda f(x)}\approx e^{-\lambda f(x_0)},\qquad \lambda\to\infty$$ where I assume that $x_0$ is a unique minimum of the function $f(x)$. These types of computations are omnipresent in physics, for example. I am interested if there are examples in the reverse direction, when finding the minimum of $f(x)$ directly is challenging, but computing something like $$\frac{\int e^{-\lambda f(x)}x}{\int e^{-\lambda f(x)}}\approx x_0,\qquad \lambda\to\infty$$ is tractable and allows to locate the minimum. Perhaps generalization to many variables or functional integrals would be more appropriate.

I hope the question is meaningful despite the lack of precision and rigor. Any comments are welcome!

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  • $\begingroup$ I don't think this "reverse engineering" will work; take the simple example $\int_{-1}^{1}e^{ix\sin\pi t}\,dt=2J_0(|x|)$, with saddle point at $t=1/2$. How would you recover that from an integral? $\endgroup$
    – Carlo Beenakker
    Commented Jan 19, 2022 at 11:55
  • $\begingroup$ I didn't check carefully but it must be that (note additional $t$ in the numerator) $\int_{-1}^1 dt e^{i x \sin \pi t} t/\int_{-1}^1 dt e^{i x \sin \pi t}$ tends to $\frac12$ in the large $x$ limit. In this case of course one can find the saddle point directly. The procedure should always work (formally), my question is if there are any examples when this is useful. $\endgroup$
    – Weather Report
    Commented Jan 19, 2022 at 11:58
  • $\begingroup$ @CarloBeenakker I'm not sure why you say the integrals do not converge. Mathematica's With[{x = 10}, NIntegrate[Exp[ x Sin[\[Pi] t]] t, {t, -1, 1}]/ NIntegrate[Exp[ x Sin[\[Pi] t]], {t, -1, 1}]] gives me 0.499989. $\endgroup$
    – Weather Report
    Commented Jan 19, 2022 at 12:41
  • $\begingroup$ I meant no convergence in the limit $x\rightarrow\infty$; I worked it out a bit more in the answer box, that seemed easier than continuing as comments. $\endgroup$
    – Carlo Beenakker
    Commented Jan 19, 2022 at 13:16
  • $\begingroup$ This trick is common in machine learning, where $\frac{\int e^{\lambda f(x)} x}{\int e^{\lambda f(x)}}$ is known as the "softmax" function, typically in the context where the integrals are finite sums. These are often easier to compute numerically because for finite $\lambda$ the softmax function is differentiable. $\endgroup$
    – Alf
    Commented Jan 19, 2022 at 17:59

2 Answers 2

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This is not an answer, but continuing the discussion in the comment box is a bit cumbersome. The point I want to make is to see if this would work for a simple test case, $f(t)=ix\sin\pi t$, with saddle point at $t=1/2$. So the question is whether we can "reverse engineer" this value of $t$ from integrals of the form $\int e^{ix\sin\pi t}dt$. The OP suggest to look at the ratio $$I(x)=\frac{\int_{-1}^1 e^{ix\sin\pi t}t\,dt}{i\int_{-1}^1 e^{ix\sin\pi t}\,dt}=\frac{1}{J_0(|x|)}\int_{0}^1 \sin(x\sin\pi t)\,t\,dt.$$ Can we somehow recover the value $t=1/2$ from the large-$x$ behaviour of $I(x)$?

From the plot of $I(x)$ it seems that this function does not converge to a definite limit as $x\rightarrow\infty$.

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    $\begingroup$ Replacing $i x$ by $x$ seems to fix the issue. $\endgroup$
    – Weather Report
    Commented Jan 19, 2022 at 13:19
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    $\begingroup$ very good, indeed $I(-ix)\rightarrow 1/2$ for $x\rightarrow\infty$, I'm just wondering how we would know the replacement $x\mapsto ix$ if all information we had came to us via integrals, but this may well be the way to go. $\endgroup$
    – Carlo Beenakker
    Commented Jan 19, 2022 at 13:28
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    $\begingroup$ Well for real exponent the integral clearly localizes near the minimum. For complex exponent, as in you original computation, both critical points $t=\pm\frac12$ probably contribute and the analysis is more involved. $\endgroup$
    – Weather Report
    Commented Jan 19, 2022 at 13:31
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This is a very particular instance of what's known as moment problem, or sometimes "inverse moment problem". One has an unknown measure $\mu$ in some space of measures, and one tries to reconstruct $\mu$ from knowlegde of $\mu_k:=\int f_k(x) d\mu(x)$, $k=0,1,\dots$, for a well-behaved sequence of functions $f_k$.

This readily generalises to several dimensions.

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  • $\begingroup$ I think the difference with my question is that I assume that the measure $\mu$ is known, but it's difficult to find its critical points directly. $\endgroup$
    – Weather Report
    Commented Jan 19, 2022 at 18:53
  • $\begingroup$ You are, essentially, trying to find the best Dirac measure appoximating your measure. $\endgroup$
    – Dima Pasechnik
    Commented Jan 19, 2022 at 21:08
  • $\begingroup$ OK, but I do not assume that the moments are given. Rather, I'm interested to look at the situation when the measure is known explicitly, it's critical points are hard to find, but the moments can still be computed. $\endgroup$
    – Weather Report
    Commented Jan 20, 2022 at 5:39
  • $\begingroup$ all the moments are seldom known in such a settting. Sometimes one can say something about a moment generating function though, and derive something from the latter. $\endgroup$
    – Dima Pasechnik
    Commented Jan 20, 2022 at 11:03

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