Can somebody help me get started with the following problem? I want to solve:
$$\theta^* := \arg\min_{\theta>0} \quad \big\| x - \sum_{n=1}^N c_{n} \bf{k}_{n}(\theta) \big\|_{2}^{2}$$
where $x$ and $\bf{k}_{n}(\theta)$ are vectors in $\mathbb{C}^L$ such that ${\bf{k}_n=\{\bf{k}_1,\bf{k}_2,...\bf{k}_N \}}$ for a given scaling factor $\theta$. $\bf{k}_{n}(\theta)$ are the time-shifts of a Gaussian, i.e.,generated by translating a Gaussian function $g(t) \in {L^2(R)} $. For any $\theta >0$ and translation $n$, $\bf{k}_{n}(\theta)$ is defined as
$$k_{n}(\theta)=g \left({t-n}\right)$$
where
$$g(t)=e^{-(t/\theta)^2}$$