## 6 Results
In addition, the influence of these variables on reconstructed image quality will be discussed, and PSNR and SSIM score values are calculated for each simulation instance; we kindly refer the reader to later discussions about reconstruction quality in this paper for more information on these scores. A number of statistical methods have been applied and results are examined to describe the convergence speed behavior of each algorithm when applied to the inverse problem of DOT. Image reconstruction in optical tomography is an ill-posed nonlinear inverse problem, the algorithms based on gradient descent present no guarantee to converge to the global minima when there are local minima in the optimization problem at hand, the convergence point depends heavily on the choice of the starting point of the optimization, and, generally, these algorithms converge (depending also on the learning rate) to the nearest local minima to the initial starting point. In this section, we address the optimization problem (image reconstruction) from the perspective of the speed of convergence (as one of the very important matters in practical use of DOT in clinical applications) rather than sensitivity of the algorithms to the choice of the initial guess with respect to their efficiency to find global minima (which is the other important practical issue in applying DOT); this last perspective is equally relevant and without a doubt needs particular attention and further analysis, but in the scope of our current paper remains an open question to follow up, as our randomized simulation design was focused on controlling the factors that influence speed of convergence. We can use the same approach as in this work to quantify (statistically speaking) the efficiency and sensitivity to reach the global minima depending on problem factors, but this obviously needs to redesign the simulation to generate the appropriate data suitable for this substantially different analysis objective. The \"blindness\" toward the globality/locality character of the reached optimum for the gradient descent-based algorithms is an inherent property because the gradient is a local concept, and by itself carries only local information about the objective function which makes these algorithms very sensitive to the choice of learning rate and initialization. The adaptive moment included features does not add to the picture but some amount of \"memory\" of the recent gradients. A rough observation that can be mentioned here is the fact that, in our generated sample data, most of the time the convergent instances for Nadam and Adam were to the global minima, but we can't really draw any statistical evidence from this naive observation because our randomized simulation design does not support this analysis. \begin{table}
\begin{tabular}{l l l l} \hline \hline  & **Adam** & **Nadam** & **AmsGrad** \\ \hline \(\beta\) & \((1.17,<2\times 10^{-16})\) & \((1.22,<2\times 10^{-16})\) & \((-0.35,<1.5\times 10^{-5})\) \\ \hline \(n\) & \((-0.005,0.6)\) & \((-0.007,0.44)\) & \((-0.044,<7.6\times 10^{-16})\) \\ \hline \(\delta\) & \((0.07,0.6)\) & \((0.1,0.5)\) & \((-0.016,0.91)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Logistic regression coefficients and corresponding \(p\)-values. First of all, we check the distributions of number of iterations (speed) for normality, in the hope to be able to harness the large and powerful available parametric statistical approaches, from literature heavily relying on this (workhorse) normal distribution. Probability distributions of speed of convergence and the log of speed of convergence are shown in the QQ plot described in Figure 2a,b, respectively. From these two graphs, it clearly appears that these distributions are very far from being reasonably considered normally or log-normally distributed. This is not a surprising fact indeed, knowing that these distributions are not symmetric to begin with, and look (strongly) skewed, but we wanted to exclude the possibilities of any approximate (left truncated) normal distributions.