# Improving Re-Identification by Estimating and Utilizing Diverse Uncertainty Types for Embeddings
## 4 Experiments
### Simple Score-Level Fusion
#### 4.3.4 Utilizing the Estimated Model Uncertainty to Adjust Feature Vectors
Also, if we optimize \(c\) specifically for a single trained model, we get a similar observation. We hypothesize that by fulfilling Equation (33), we may be able to derive a formula to compute \(c\). First, we can write the uncertainty vectors in the detailed version, to get Equation (34). Now it is more easily observable that the square root is applied element-wise, as mentioned above. \[f\Big{(}\sqrt{\Sigma^{(M)}}\Big{)} =2\cdot f\Big{(}\Sigma^{(M)}\Big{)} \tag{33}\] \[f\Big{(}\bigg{[}\sqrt{\Sigma_{1}^{(M)}}\;\sqrt{\Sigma_{2}^{(M)} }\;\;\ldots\;\sqrt{\Sigma_{d}^{(M)}}\bigg{]}\Big{)} =2\cdot f\Big{(}\Big{[}\Sigma_{1}^{(M)}\;\Sigma_{2}^{(M)}\;\ldots \;\Sigma_{d}^{(M)}\Big{]}\Big{)} \tag{34}\]  
A function that solves Equation (34) is shown in Equation (35). \[f\Big{(}\Sigma^{(M)}\Big{)}=\frac{\lambda}{\left|\ln\left(\prod\limits_{i=1}^{ d}\Sigma_{i}^{(M)}\right)\right|} \tag{35}\]  
The mathematical proof that the function described in Equation (35) does fulfill Equations (33) and (34), is given in Equations (36)-(43). \[f\Big{(}\sqrt{\Sigma^{(M)}}\Big{)} =f\Big{(}\bigg{[}\sqrt{\Sigma_{1}^{(M)}}\;\sqrt{\Sigma_{2}^{(M)} }\;\;\ldots\;\sqrt{\Sigma_{d}^{(M)}}\bigg{]}\Big{)} \tag{36}\] \[=\frac{\lambda}{\left|\ln\left(\prod\limits_{i=1}^{d}\sqrt{\Sigma _{i}^{(M)}}\right)\right|}\] (37) \[=\frac{\lambda}{\left|\ln\left(\prod\limits_{i=1}^{d}\left(\Sigma _{i}^{(M)}\right)^{0.5}\right)\right|}\] (38) \[=\frac{\lambda}{\left|\ln\left(\left(\prod\limits_{i=1}^{d} \Sigma_{i}^{(M)}\right)^{0.5}\right)\right|}\] (39) \[=\frac{\lambda}{\left|0.5\cdot\ln\left(\prod\limits_{i=1}^{d} \Sigma_{i}^{(M)}\right)\right|}\] (40) \[=\frac{\lambda}{0.5\cdot\left|\ln\left(\prod\limits_{i=1}^{d} \Sigma_{i}^{(M)}\right)\right|}\] (41) \[f\Big{(}\sqrt{\Sigma^{(M)}}\Big{)} =2\cdot f\Big{(}\Sigma^{(M)}\Big{)} \tag{33}\]0 (42) \[f\Big{(}\sqrt{\Sigma^{(M)}}\Big{)} =2\cdot f\Big{(}\Sigma^{(M)}\Big{)} \tag{33}\]1Now, we can reformulate Equation (35) as a function of a norm of an uncertainty vector (Equations (44)-(47)):  
\[f\Big{(}\sqrt{\Sigma^{(M)}}\Big{)} =2\cdot f\Big{(}\Sigma^{(M)}\Big{)} \tag{33}\]2 \[f\Big{(}\sqrt{\Sigma^{(M)}}\Big{)} =2\cdot f\Big{(}\Sigma^{(M)}\Big{)} \tag{33}\]3 (45) \[ \begin{cases}=\frac{\lambda}{\sum\limits_{i=1}^{d}\left|\ln\left( \Sigma_{i}^{(M)}\right)\right|}&\text{if }\Sigma_{i}^{(M)}<1\ \forall\ i\in\{1,2,\ldots,d\}\\ \approx\frac{\lambda}{\sum\limits_{i=1}^{d}\left|\ln\left(\Sigma_{i}^{(M)} \right)\right|}&\text{otherwise}\ \ \ \text{(in 2.88\% of uncertainty vectors)}\end{cases} \] (46) \[ \begin{cases}=\frac{\lambda}{\left\|\ln\left(\Sigma^{(M)}\right) \right\|_{1}}&\text{if }\Sigma_{i}^{(M)}<1\ \forall\ i\in\{1,2,\ldots,d\}\\ \approx\frac{\lambda}{\left\|\ln\left(\Sigma^{(M)}\right)\right\|_{1}}&\text{ otherwise}\ \ \ \text{(in 2.88\% of uncertainty vectors)}\end{cases}  \tag{47}\]  
Note, that the derived formula using the norm (Equation (47)) is only equivalent to Equation (44) if all elements of the vector are smaller than 1. This requirement is fulfilled in the majority of cases. However, in 2.88% of the uncertainty vectors, at least one element is larger than 1. For these vectors, we performed a worst-case analysis, that showed, that the influence of the large values for some elements of the vector is marginal. Equation (48) describes the extent to which the two computations deviate in cases where \(\Sigma_{i}^{(M)}>1\) for some \(i\). For the vector with the largest deviation over all runs, we observed a factor \(\alpha=1.0024\). This means that the error never exceeds 0.24%. This minimal error is negligible given that the achieved performance, as measured by mAP, is nearly unaffected when varying \(c\) by this amount. \[\frac{\lambda}{\left\|\ln\left(\Sigma^{(M)}\right)\right\|_{1}}\leq\frac{ \lambda}{\left|\sum\limits_{i=1}^{d}\ln\left(\Sigma_{i}^{(M)}\right)\right|}< \alpha\cdot\frac{\lambda}{\left\|\ln\left(\Sigma^{(M)}\right)\right\|_{1}} \tag{48}\]  
We conclude that we can use Equation (47) to compute \(c\). Unfortunately, in Equation (47) a new parameter \(\lambda\) is introduced.