A maximization problem in functional analysis and data

Consider the minimization problem described this paper. Let $$f_{\lambda}$$ be the minimizer. As a part of extending my work, I am able to show the following facts

$$\lim_\limits{\lambda \to 0}\|f_{\lambda}\|_{L^2} = 0$$ and $$\lim_\limits{\lambda \to \infty}\|f_{\lambda}\|_{L^2} = 0$$

My problem now is (as I would like to extend my work), find $$\lambda \in (0,\infty)$$ for which $$\|f_{\lambda}\|_{L^2}$$ is maximum. Appreciate your suggestions to solve this problem.

The minimization problem from the linked paper is given below for the self containment of the post. If given that $$k>\frac{m}{2}$$, the paper proves that there is a unique minimizer for the functional $$C(f)$$ in the set $$S$$.

It is given that $$k>\frac{m}{2}$$

My progress

Partial Progress :

Progress : I am able to derive the corresponding PDE equations for the problem.

Let $$f(\lambda,x) = f_{\lambda}(x)$$. Then

The first equation corresponds to maximizing $$\|f_{\lambda}\|$$, while the second PDE is for the minimization problem associated with the parameter $$\lambda$$.

The second equation (minimization problem), given any $$\lambda$$, I can solve for $$f(\lambda,.)$$ either using linear algebra or steepest descent algorithm, which I have described in my article. Now I need to use this solution and the first equation to obtain $$\lambda$$, which is a problem I am facing.

Trying to solve using linear algebra, by formulating the discrete version of the problem using Fourier series coefficients and Plancheral theorem, I get stuck at a matrix problem.

More Partial Progress

An Iterative algorithm which is a modified steepest descent.

1. Initialize $$f$$.

2. Assuming some $$\lambda$$ and assuming gradient of $$C_{\lambda}(f)$$ wrt $$f$$ be $$\nabla_f C_{\lambda}(f)$$, and if we were to update $$f$$ with this gradient as in we do in steepest descent, it would be $$f^u_\lambda = f - \delta \nabla_f C_{\lambda}(f)$$, where $$\delta$$ is a constant learning rate. Now set $$\frac{\partial\|f^u_\lambda\|}{\partial \lambda} = 0$$ and solve for $$\lambda$$. Let the root be $$\lambda_0$$.

3. Update $$f = f^u_{\lambda_0}$$. (update $$f$$ as in steepest descent, but using $$\lambda$$ value as $$\lambda_0$$ which was computed in step 2.)

4. check some convergence criterion and if not met, go to step 2.

I have implemented this numerically and it converges as desired. Need to work on the proof.

PS : This was first posted on MO by me, 3 months back. Link

• added progress to the question itself, rather than a separate answer. Jul 29 '19 at 9:56
• This was first posted on MO by me, 3 months back. Link : mathoverflow.net/q/332439/14414 Jul 29 '19 at 13:53
• Done. @Ilmari I have removed the dead link. Oct 12 '19 at 1:59

Some more progress: I have found a direct solution to the PDE (E-L equation of the minimization problem). I hope it will help in finding the $$\lambda$$ that maximizes $$\|f_{\lambda}\|_{L^2}$$.

The solution to PDE is described below.

Let $$g_{\lambda}(\boldsymbol{x}) = \sum\limits_{{\pmb{\eta}\in\mathbb{Z}^m}}\frac{1}{1+\lambda\sum\limits_{i=1}^{m}\eta_i^{2k}} \cos({2\pi \pmb{\eta}\cdot\pmb{x}})$$

Assuming $$k >\frac{m}{2}$$, using Bochner's theorem we can see that the function $$g_{\lambda}$$ is positive definite.

The solution to the PDE (the Euler-Lagrange equation for the minimization problem) is then given as

$$f_{\lambda}(\pmb{x}) = \sum\limits_{i=1}^{n}c_ig_{\lambda}(\pmb{x-p_i})$$ where $$c_i \in \mathbb{R}$$.

Let $$\pmb{c} = [c_1,c_2,...c_n]^T$$.

We can determine $$\pmb{c}$$ by substituting the above expression for $$f_{\lambda}(\pmb{x})$$ in the PDE equation and is given as $$\pmb{c} = (G_{\lambda}+I)^{-1}L$$ where the matrix $$G_{\lambda}$$ is given as $$G_{\lambda} = [\gamma_{ij}]_{n\times n},\gamma_{ij} = g_{\lambda}(\pmb{p_i}-\pmb{p_j})$$ and $$L = [a_1,a_2,....a_n]^T$$

The matrix $$G_{\lambda}$$ is positive semidefinite as $$g_{\lambda}$$ is a positive definite function. Hence the matrix $$G_{\lambda}+I$$ is positive definite and is invertible.

An Expression for $$\|f_{\lambda}\|$$

We derive an expression for $$\|f_{\lambda}\|$$. Let $$$$z_{\lambda}(\boldsymbol{x}) = \sum\limits_{{\pmb{\eta}\in\mathbb{Z}^m}}\frac{1}{(1+\lambda\sum\limits_{i=1}^{m}\eta_i^{2k})^2} \cos({2\pi \pmb{\eta}\cdot\pmb{x}})$$$$

Define the matrix $$Z_{\lambda} = [\beta_{i,j}]_{n\times n}$$ where $$\beta_{i,j} = z_{\lambda}(\boldsymbol{p_i}-\boldsymbol{p_j})$$ Using Parseval's theorem and some algebraic manipulation, we can show that $$\|f_{\lambda}\| = \pmb{c}^T Z_{\lambda} \pmb{c}$$ and there by $$\|f_{\lambda}\| = ((G_{\lambda}+I)^{-1}L)^T Z_{\lambda} (G_{\lambda}+I)^{-1}L$$ and hence $$\|f_{\lambda}\| = L^T(G_{\lambda}+I)^{-1} Z_{\lambda} (G_{\lambda}+I)^{-1}L$$ Denoting $$K_{\lambda} = (G_{\lambda}+I)^{-1}$$ one can write $$\|f_{\lambda}\| = L^T K_{\lambda}Z_{\lambda}K_{\lambda}L$$

To determine $$\lambda_0$$, the maxima of $$\|f_{\lambda}\|$$, taking derivative with repect to $$\lambda$$ and setting it to zero, we get $$\frac{\partial{\|f_{\lambda}\|}}{\partial{\lambda}} = L^T\frac{\partial{(K_{\lambda}Z_{\lambda}K_{\lambda})}}{\partial{\lambda}}L = 0$$