I am trying to evaluate the following minimize-sum-square optimization expression:
$$\min_{a\in A}\sum_{x\in X}(\int_{-\infty}^{\infty} e^{-itx}\phi(t, a) dt - \int_{-\infty}^{\infty} e^{-itx}\phi(t, b) dt)^2$$
whereas A is a set of parameters, b is one element of parameters outside this set of parameters A and is fixed. X is set of arbitrary float numbers.
To give a bit of context: This problem I am trying to solve describes a problem to fit the distribution of a characteristic function to another one that is fixed.
I tried to solve this via scipy optimize with X being equally linearly spaced between -1 and 1 for 100 datapoints. The results have been quite instable, small changes to the input b gives often big rapid changes in a.
I am trying to simplify this term:
$$\sum_{x\in X}(\int_{-\infty}^{\infty} e^{-itx}\phi(t, a) dt - \int_{-\infty}^{\infty} e^{-itx}\phi(t, b) dt)^2$$
$$\sum_{x\in X}(\int_{-\infty}^{\infty} e^{-itx}(\phi(t, a)-\phi(t, b)) dt )^2$$
By the Hoelder Inequation, the following holds
$$\sum_{x\in X}(\int_{-\infty}^{\infty} e^{-itx}(\phi(t, a)-\phi(t, b)) dt)^2 \leq \sum_{x\in X}\int_{-\infty}^{\infty} (e^{-itx}(\phi(t, a)-\phi(t, b)))^2 dt$$
$$\leq \sum_{x\in X}\int_{-\infty}^{\infty} e^{-2itx}(\phi(t, a)-\phi(t, b))^2 dt$$
$$\leq \int_{-\infty}^{\infty} (\phi(t, a)-\phi(t, b))^2 \sum_{x\in X} e^{-2itx} dt$$
So the minimization problem can be expressed as:
$$\min_{a\in A}\int_{-\infty}^{\infty} (\phi(t, a)-\phi(t, b))^2 \sum_{x\in X} e^{-2itx} dt$$
Trying this, I am unfortunately getting non-sense results, I am wondering the following:
- This is transformation of the expression well done? Can I apply the Hoelder Inequalition?
- I was thinking, since $\sum_{x\in X} e^{-2itx}$ is always $>0$, assuming 1. is correctly applied, since it is the unity circle * 2 and the minimum of the problem must be 0, can we only minimize $$\min_{a\in A}\int_{-\infty}^{\infty} (\phi(t, a)-\phi(t, b))^2 dt$$
- I have read about variational calculus but am unfortunately not specialized in that area (never had it in uni). Can this be applied to this problem? If yes, is there literature to this?
Thanks!