I need your expertise in solving the following problem:
Given $x \in \mathbb{R}^d$ and let $A \in \mathbb{R}^{d \times d}$ a vector and and an inverse matrix respectively. Let $\alpha \in \mathbb{R}^+$.
Is there a way to prove that: $$ e^{-\left\| A\right\|_2^2 + 2 \cdot \left\| A^{-1}x\right\|_1 - \left\| x\right\|_2^2 } \leq \alpha \cdot \sum\limits_{i=1}^d e^{-\left\| Ae_j\right\|_2^2 + 2\cdot \left| A^{-1}xe_j\right| - \left\| x\right\|_2^2 }$$
And if so how can we bound $\alpha$?
Please advise and Thanks in advance.