# Deep Neural Networks Classification via Binary Error-Detecting Output Codes
## 3 Results and Discussion
### Performance on CIFAR-10
Decision and training based on Euclid distance improve output values, on average, and we also see average performance results on Hadamard codewords despite using 15 bit-long codewords. According to Zadeh fuzzy logic, the decision tends to decrease the maximum output error, given the class codeword during the training. During the test, the output is assigned to the codeword with a minimum of maximal bit errors. Loosely speaking, while a one-hot softmax pulls the best bit value to \"1\", the proposed [PERSON]'s decision pushes the worst error to \"0\". Table 4 shows similar results for both strategies. We can conclude that what the softmax gives to one-hot encoding, the Zadeh fuzzy logic provides to binary coding. In the case of the [PERSON] decision, we can offer more codewords to the referee. It decreases the FCA compared with the accuracy, and the difference indicates how many samples were assigned to the codewords outside the class-codewords. We interpret the case as error detection, and it bypasses classification. If only class-codewords are applied during the testing, there is no difference between the accuracy and FCA. It is the case of one-hot encoding with softmax where only class-codewords can be used. Therefore, if the classification returns one of the class-codewords, it has to be accepted. The misclassified codeword causes an undetectable error. Table 4 indicates some benefits of error detection. The exclusion of untrustworthy samples increases the classification reliability. A well-known way to increase reliability is to introduce a level of decision-making. All decision rules are based on the evaluation of a quantitative variable. The usual way to improve the decision reliability is to divide the variable interval by threshold \(\Delta\) and reject all samples under the threshold, bypassing the decision procedure. This rule can also be applied to the softmax concept in which the variable is in a range \([1/n,\ 1]\), where \(n\) is the number of classes. Therefore, the threshold setting below \(1/n\) has no meaning, and it gives no improvement. Simply put, in the softmax decision, the sample is accepted for classification if the highest value of the output exceeds the given threshold. We can apply the same approach in the Zadeh decision and decide if \(max\big{\{}S^{0},\ldots,S^{w}\big{\}}\geq\Delta\). Theorem 1 says that the threshold \(\Delta<0.5\) brings no improvement in the full codespace decision. We show the accuracy-reliability characteristic using different thresholds for proposed CRC7 with the Zadeh decision and baseline one-hot encoding with softmax in Figure 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Network** & **Code** & **Decision** & **Accuracy** & **FCA** & **Undetectable Error** & **Reliability** \\ \hline CNN1 & CRC7 & Zadeh & **92.44** & 92.01 & **6.77** & **93.15** \\ CNN1 & one-hot & Zadeh & 92.01 & 91.56 & 7.29 & 92.63 \\ CNN1 & one-hot & softmax & 92.15 & **92.15** & 7.85 & 92.15 \\ CNN1 & CRC7 & softmax & 91.85 & 91.85 & 8.15 & 91.85 \\ CNN1 & CRC7 & bit threshold & 89.56 & 89.56 & 6.8 & 92.94 \\ CNN1 & Hadamard & Euclid & 89.92 & 88.6 & 8.2 & 91.53 \\ CNN2 & CRC7 & Zadeh & 93.97 & 93.47 & **5.44** & **94.5** \\ CNN2 & one-hot & Zadeh & 93.4 & 92.66 & 5.76 & 94.14 \\ CNN2 & one-hot & softmax & **94.07** & **94.07** & 5.93 & 94.07 \\ CNN2 & CRC7 & softmax & 93.88 & 93.88 & 6.12 & 93.88 \\ ResNet20v2 & CRC7 & Zadeh & 91.56 & 91.27 & **7.97** & **91.97** \\ ResNet20v2 & one-hot & Zadeh & 84.16 & 83.28 & 14.5 & 85.17 \\ ResNet20v2 & one-hot & softmax & **91.67** & **91.67** & 8.33 & 91.67 \\ ResNet20v2 & CRC7 & softmax & 91.34 & 91.34 & 8.66 & 91.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance of the various output coding and decision strategies. Figure 3 maps values from Table 4 to the left located endpoints of the curves, where no threshold is applied. As the threshold increases, the points are shifted to the right. The error detection property makes the location of the zero threshold points outside the line \(y=x\).