I've been trying to understand Probabilistic PCA and I got stuck on the following equation that tries to find what should the discarded eigenvalues be
$$\mathcal{L} = -\frac{N}{2}\left\{d \ln(2\pi)+\sum_{j=1}^{q'}\ln\lambda_j+(d-q')ln\left(\frac{1}{d-q'}\sum_{j=q'+1}^{d}\lambda_j\right)+d \right\}$$
The authors argue that minimizing the above equation is same as minimizing
$$E := \ln\left(\frac{1}{d-q'}\sum_{j=q'+1}^{d}\lambda_j\right) - \frac{1}{d-q'}\sum_{j=q'+1}^{d}\ln\lambda_j$$
After this they say
Interestingly, the minimization of E leads only to the requirement that the discarded $\lambda_j$ be adjacent within the spectrum of the ordered eigenvalues of S
I'm unable to prove the above statement. I got some intuition from the following equivalent problem
$$E = \ln{x^Te} - \frac{1}{n}\sum_{i=1}^{n}\ln x_i$$
$$\nabla_x E = \frac{e}{x^Te} - \frac{1}{n}\sum_{i=1}^{n}\frac{e_i}{e_i^Tx}$$
A trivial solution occurs when all the components of $x$ are equal.
I'm unable to however convert this intuition into a rigorous solution, further is the trivial solution the only critical point?