# Deep Error-Correcting Output Codes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- We integrate an online learning method into the ECOC coding, which improves the efficiency of ECOC, especially for large-scale applications.
- We employ ECOC as building blocks of deep networks, which sufficiently utilizes the available label information of data and improves the effectiveness and efficiency of previous deep learning algorithms.
- We propose the DeepECOCs model, which combines the ideas of ensemble learning, online learning and deep learning.

## 2. Related Work

## 3. Deep Error-Correcting Output Codes (DeepECOCs)

#### 3.1. The ECOC Framework

#### 3.2. Incremental Support Vector Machines (Incremental SVMs)

#### 3.2.1. KKT Conditions

#### 3.2.2. Incremental Learning Procedure

- (1)
- Initialize ${\alpha}_{a}$ to 0, then calculate ${g}_{a}$;
- (2)
- If ${g}_{a}>0$, terminate (a is not a margin or error vector);
- (3)
- If ${g}_{a}\le 0$, apply the largest possible increment ${\alpha}_{a}$ so that one of the following conditions occurs:
- (a)
- ${g}_{a}=0$: add a to margin set $\mathcal{S}$, terminate;
- (b)
- ${\alpha}_{a}=C$: add a to error set $\mathcal{E}$, terminate;
- (c)
- Elements of ${X}^{l+1}$ migrate across $\mathcal{S}$, $\mathcal{E}$ and $\mathcal{R}$; update membership of elements, and if $\mathcal{S}$ changes, update $\mathcal{R}$ accordingly.

#### 3.3. DeepECOCs

Algorithm 1 The training procedure of DeepECOCs; L is the number of layers, I-SVM($\mathbf{x},\mathbf{y}$) is the incremental SVM binary classifier, $s\left(x\right)$ is the sigmoid function, and $\mathrm{softmax}\left(\mathbf{x}\right)$ is the softmax function. |

Require: |

The set of training samples $\mathbf{X}=\{{\mathbf{x}}_{1},\dots ,{\mathbf{x}}_{j},\dots ,{\mathbf{x}}_{n}\}$; |

The labels corresponding to training samples $\mathbf{y}=\{{y}_{1},\dots ,{y}_{j},\dots ,{y}_{n}\}$. |

Ensure: |

Parameters $\mathbf{W}$ and $\mathbf{b}$. |

1: Initialize the first layer input ${\mathbf{Z}}^{1}=\mathbf{X}$; |

2: for $i=1$ to $L-1$ do |

3: Initialize the weights and bias of i-th ECOC module; |

4: ${\mathbf{W}}^{i}=\mathbf{0},{\mathbf{b}}^{i}=\mathbf{0}$; |

5: Pre-train process (ECOC coding step): |

6: (1) Learn the ECOC matrix with a coding strategy, and obtain $\mathbf{M}\in {\{-1,1\}}^{C\times P}$ (binary case) or $\mathbf{M}\in {\{-1,0,1\}}^{C\times P}$ (ternary case); |

7: (2) Train the incremental SVM binary classifiers according to $\mathbf{M}$: |

8: for $k=1$ to P do |

9: $({\alpha}_{k},{b}_{k}^{i})\leftarrow $ I-SVM$({\mathbf{Z}}^{i},\mathbf{y})$; |

10: ${\mathbf{w}}_{k}^{i}={\sum}_{j=1}^{N}{\alpha}_{j}{y}_{j}{\mathbf{Z}}_{j}^{i}$; |

11: end for |

12: ${\mathbf{W}}^{i}=\{{\mathbf{w}}_{1}^{i},\dots ,{\mathbf{w}}_{k}^{i},\dots ,{\mathbf{w}}_{P}^{i}\}$; |

13: ${\mathbf{b}}^{i}=\{{b}_{1}^{i},\dots ,{b}_{k}^{i},\dots ,{b}_{P}^{i}\}$; |

14: ${\mathbf{z}}^{i+1}=s({\left({\mathbf{W}}^{i}\right)}^{T}{\mathbf{z}}^{i}+{\mathbf{b}}^{i})$; |

15: end for |

16: ${\mathbf{z}}^{L}=\mathrm{softmax}\left({\mathbf{z}}^{L-1}\right)$. |

17: Use the back-propagation algorithm for fine-tuning. |

18: return $\mathbf{W}=\left\{{\mathbf{W}}^{i}\right\}$ and $\mathbf{b}=\left\{{\mathbf{b}}^{i}\right\}$, $i=1,\dots ,L-1$. |

#### 3.4. Combining Convolutional Neural Networks (CNNs) with DeepECOCs

## 4. Experiments

#### 4.1. Classification on 16 UCI Data Sets

#### 4.2. Classification on the USPS Data Set

#### 4.3. Classification on the MNIST Data Set

#### 4.4. Classification on the CMU Mocap Data Set

#### 4.5. Classification on the CIFAR-10 Data set

#### 4.6. Combining CNNs with DeepECOCs

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**6 coding matrices encoded with different coding strategies. (

**a**) one-versus-one. (

**b**) one-versus-all. (

**c**) DECOC. (

**d**) ECOCONE. (

**e**) Dense random. (

**f**) Sparse random.

**Figure 2.**A deep architecture combining CNNs and DeepECOCs. The CNN module produces a vector presentation of data as shown with the long black bar. DeepECOC takes this data representation as input and performs the classification task. This deep architecture can be trained in an end-to-end manner.

**Figure 3.**Classification accuracy on the USPS data set. ECOCONE(1)∼ECOCONE(3) are the different initial coding strategies used in ECOCONE for DeepECOCs.

**Figure 6.**Classification accuracy on the MNIST data set. The architecture is 784-${Z}_{1}$-${Z}_{2}$-${Z}_{3}$-10.

Problem | ♯ of I | ♯ of A | ♯ of C | Problem | ♯ of I | ♯ of A | ♯ of C |
---|---|---|---|---|---|---|---|

Dermatology | 366 | 34 | 6 | Yeast | 1484 | 8 | 10 |

Iris | 150 | 4 | 3 | Satimage | 6435 | 36 | 7 |

Ecoli | 336 | 8 | 8 | Letter | 20,000 | 16 | 26 |

Wine | 178 | 13 | 3 | Pendigits | 10,992 | 16 | 10 |

Glass | 214 | 9 | 7 | Segmentation | 2310 | 19 | 7 |

Thyroid | 215 | 5 | 3 | Optdigits | 5620 | 64 | 10 |

Vowel | 990 | 10 | 11 | Shuttle | 14,500 | 9 | 7 |

Balance | 625 | 4 | 3 | Vehicle | 846 | 18 | 4 |

**Table 2.**Classification accuracy on the 16 UCI data sets obtained by comparing LibSVM and incremental SVM. The best results for each scenario are highlighted in bold face.

Problem | OneVsOne | OneVsAll | DECOC | ECOCONE | ||||
---|---|---|---|---|---|---|---|---|

LibSVM | I-SVM | LibSVM | I-SVM | LibSVM | I-SVM | LibSVM | I-SVM | |

Dermatology | 0.9671 | 0.9770 | 0.7928 | 0.9605 | 0.9671 | 0.9638 | 0.9671 | 0.9770 |

Iris | 0.9333 | 0.9333 | 0.7030 | 0.7030 | 0.9333 | 0.9333 | 0.9333 | 0.9333 |

Ecoli | 0.5944 | 0.6507 | 0.3940 | 0.4553 | 0.5281 | 0.5828 | 0.5944 | 0.6556 |

Wine | 0.9892 | 0.9892 | 0.9731 | 0.9731 | 0.9892 | 0.9731 | 0.9839 | 0.9892 |

Glass | 0.4838 | 0.4189 | 0.3216 | 0.3270 | 0.2973 | 0.3486 | 0.3027 | 0.5378 |

Thyroid | 0.5897 | 0.6239 | 0.5897 | 0.6239 | 0.5897 | 0.4017 | 0.5897 | 0.6239 |

Vowel | 0.4187 | 0.4956 | 0.3953 | 0.6591 | 0.2736 | 0.3278 | 0.4476 | 0.4956 |

Balance | 0.8927 | 0.8927 | 0.9042 | 0.9042 | 0.8927 | 0.8927 | 0.8927 | 0.8927 |

Yeast | 0.4741 | 0.5744 | 0.2383 | 0.2760 | 0.3751 | 0.5693 | 0.4147 | 0.6000 |

Satimage | 0.8361 | 0.8692 | 0.7620 | 0.8334 | 0.7411 | 0.8532 | 0.8305 | 0.8604 |

Letter | 0.7443 | 0.8216 | 0.2166 | 0.4536 | 0.7530 | 0.8375 | 0.7440 | 0.8120 |

Pendigits | 0.9688 | 0.9859 | 0.9021 | 0.9773 | 0.9065 | 0.9758 | 0.9691 | 0.9850 |

Segmentation | 0.8595 | 0.8872 | 0.6187 | 0.6759 | 0.7969 | 0.6706 | 0.8601 | 0.8872 |

OptDigits | 0.9492 | 0.9860 | 0.8866 | 0.9649 | 0.8793 | 0.9808 | 0.9492 | 0.9860 |

Shuttle | 0.9178 | 0.9475 | 0.8312 | 0.9034 | 0.8352 | 0.9407 | 0.9179 | 0.9532 |

Vehicle | 0.6324 | 0.7595 | 0.7378 | 0.4892 | 0.6054 | 0.7243 | 0.6324 | 0.7595 |

Problem | $\mathit{\eta}$ | Epoch | Problem | $\mathit{\eta}$ | Epoch |
---|---|---|---|---|---|

Dermatology | 0.1 | 2000 | Yeast | 0.01 | 4000 |

Iris | 0.1 | 400 | Satimage | 0.01 | 4000 |

Ecoli | 0.1 | 2000 | Letter | 0.01 | 8000 |

Wine | 0.1 | 2000 | Pendigits | 0.01 | 2000 |

Glass | 0.01 | 4000 | Segmentation | 0.01 | 8000 |

Thyroid | 0.1 | 800 | Optdigits | 0.01 | 2000 |

Vowel | 0.1 | 4000 | Shuttle | 0.1 | 2000 |

Balance | 0.1 | 4000 | Vehicle | 0.1 | 4000 |

**Table 4.**Classification accuracy and standard deviation obtained by DeepECOCs and the compared approaches on the 16 UCI data sets. V1∼V6 represents DeepECOCs with different coding strategies, including OneVsOne, OneVsAll, DECOC and ECOCONE (initialized by OneVsOne, OneVsAll and DECOC). The best results are highlighted in boldface.

Problem | Single | AE | DAE | V1 | V2 | V3 | V4 |
---|---|---|---|---|---|---|---|

Dermatology | 0.9513 | 0.9429 ± 0.0671 | 0.9674 ± 0.0312 | 0.9731 ± 0.0314 | 0.8852 ± 0.0561 | 0.9722 ± 0.0286 | 0.8834 ± 0.0349 |

Iris | 0.9600 | 0.9600 ± 0.0562 | 0.9333 ± 0.0889 | 0.8818 ± 0.0695 | 0.8137 ± 0.0768 | 0.9667 ± 0.0471 | 0.9667 ± 0.0471 |

Ecoli | 0.8147 | 0.7725 ± 0.0608 | 0.8000 ± 0.0362 | 0.8275 ± 0.0427 | 0.7868 ± 0.0633 | 0.9102 ± 0.0701 | 0.9183 ± 0.0611 |

Wine | 0.9605 | 0.9765 ± 0.0264 | 0.9563 ± 0.0422 | 0.9063 ± 0.0793 | 0.9625 ± 0.0604 | 0.9875 ± 0.0264 | 0.9625 ± 0.0437 |

Glass | 0.6762 | 0.6669 ± 0.1032 | 0.6669 ± 0.0715 | 0.7563 ± 0.0653 | 0.6875 ± 0.0877 | 0.8013 ± 0.0476 | 0.7830 ± 0.0893 |

Thyroid | 0.9210 | 0.9513 ± 0.0614 | 0.9599 ± 0.0567 | 0.8901 ± 0.1177 | 0.9703 ± 0.0540 | 0.9647 ± 0.0431 | 0.9560 ± 0.0633 |

Vowel | 0.7177 | 0.6985 ± 0.0745 | 0.7101 ± 0.0756 | 0.7020 ± 0.0529 | 0.6563 ± 0.0721 | 0.7628 ± 0.0716 | 0.6874 ± 0.0438 |

Balance | 0.8222 | 0.8036 ± 0.0320 | 0.8268 ± 0.0548 | 0.8528 ± 0.0534 | 0.8108 ± 0.0634 | 0.8879 ± 0.1331 | 0.9090 ± 0.0438 |

Yeast | 0.5217 | 0.5641 ± 0.0346 | 0.5891 ± 0.0272 | 0.5861 ± 0.0318 | 0.5368 ± 0.0582 | 0.6080 ± 0.0385 | 0.5968 ± 0.0378 |

Satimage | 0.8537 | 0.8675 ± 0.0528 | 0.8897 ± 0.0304 | 0.8763 ± 0.0576 | 0.8238 ± 0.0453 | 0.8977 ± 0.0752 | 0.9108 ± 0.0483 |

Letter | 0.9192 | 0.9234 ± 0.0547 | 0.9381 ± 0.0641 | 0.9498 ± 0.0587 | 0.9322 ± 0.0251 | 0.9553 ± 0.0327 | 0.9465 ± 0.0414 |

Pendigits | 0.9801 | 0.9831 ± 0.0123 | 0.9886 ± 0.0034 | 0.9828 ± 0.0047 | 0.9768 ± 0.0034 | 0.9917 ± 0.0021 | 0.9817 ± 0.0082 |

Segmentation | 0.9701 | 0.9584 ± 0.0317 | 0.9596 ± 0.0211 | 0.9683 ± 0.0412 | 0.9567 ± 0.0527 | 0.9757 ± 0.0296 | 0.9724 ± 0.0371 |

Optdigits | 0.9982 | 0.9785 ± 0.0101 | 0.9856 ± 0.0088 | 0.9882 ± 0.0104 | 0.9845 ± 0.0122 | 0.9934 ± 0.0027 | 0.9901 ± 0.0033 |

Shuttle | 0.9988 | 0.9953 ± 0.0012 | 0.9976 ± 0.0014 | 0.9991 ± 0.0017 | 0.9983 ± 0.0018 | 0.9996 ± 0.0012 | 0.9993 ± 0.0010 |

Vehicle | 0.7315 | 0.6987 ± 0.0521 | 0.7348 ± 0.0454 | 0.7128 ± 0.0384 | 0.6624 ± 0.0472 | 0.7466 ± 0.0481 | 0.7097 ± 0.0521 |

Mean rank | 5.2500 | 6.3125 | 4.9375 | 4.3125 | 7.0625 | 1.4375 | 3.1875 |

**Table 5.**The classification accuracy on the CMU mocap data set. The best results for each scenario are highlighted in bold face.

Problem | AE | DAE | V1 | V2 | V4 | V5 |
---|---|---|---|---|---|---|

CMU | 0.6171 | 0.6422 | 0.8030 | 0.6364 | 0.7652 | 0.6030 |

**Table 6.**The classification accuracy on the CIFAR-10 data set. The best results for each scenario are highlighted in bold face.

Problem | AE | DAE | V1 | V2 |
---|---|---|---|---|

CIFAR(36) | 0.3501 | 0.3678 | 0.4530 | 0.3895 |

CIFAR(256) | 0.4352 | 0.4587 | 0.4936 | 0.4521 |

Problem | V3 | V4 | V5 | V6 |

CIFAR(36) | 0.4031 | 0.5089 | 0.4517 | 0.4752 |

CIFAR(256) | 0.5236 | 0.5588 | 0.4589 | 0.5224 |

**Table 7.**The classification accuracy obtained on the CIFAR-10 data set. The best result is highlighted in boldface.

Methods | ResNet-Baseline | ResNet-DeepECOCs |
---|---|---|

Accuracy | 0.9098 | 0.9208 |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.-N.; Wei, H.; Zheng, Y.; Dong, J.; Zhong, G.
Deep Error-Correcting Output Codes. *Algorithms* **2023**, *16*, 555.
https://doi.org/10.3390/a16120555

**AMA Style**

Wang L-N, Wei H, Zheng Y, Dong J, Zhong G.
Deep Error-Correcting Output Codes. *Algorithms*. 2023; 16(12):555.
https://doi.org/10.3390/a16120555

**Chicago/Turabian Style**

Wang, Li-Na, Hongxu Wei, Yuchen Zheng, Junyu Dong, and Guoqiang Zhong.
2023. "Deep Error-Correcting Output Codes" *Algorithms* 16, no. 12: 555.
https://doi.org/10.3390/a16120555