# Stable Low-Rank CP Decomposition for Compression of Convolutional Neural Networks Based on Sensitivity

^{*}

## Abstract

**:**

## 1. Introduction

- The sensitivity of each convolutional layer is computed by Hessian trace to determine the amount of redundancy of each layer, and then select the optimal ranks for CP decomposition by considering the redundancy we obtained.
- In order to overcome the CP decomposition instability problem, an efficient optimization method is developed to quantify the sensitivity of CP decomposition and add sensitivity-constrained optimization during the process.
- Another potential approach is proposed to address the problem of instability in CP decomposition by using iterative fine-tuning techniques. The layer that exhibits more sensitivity should be prioritized for compression to minimize the accuracy degradation of the compressed model.

## 2. Related Work

## 3. The Proposed Method

#### 3.1. CP Decomposition for Convolutional Layers

#### 3.2. Rank Selection Based on Sensitivity

#### 3.3. Sensitivity-Constrained Optimization in CP Decomposition

#### 3.4. Iterative Fine-Tuning Based on Sensitivity

- Estimating the sensitivity of each layer in the original model.
- Sorting each network layer according to the order of the sensitivity information content of each network layer from large to small.
- Determining the rank of each layer according to its sensitivity and an average rank we set in advance.
- Iterative compressing and fine-tuning based on the sensitivity of each layer. The sorting results obtained from the second step and the rank obtained from the third step are used to iteratively compress and fine-tune the network layer-by-layer.

## 4. Experimental Results

#### 4.1. Experimental Results on CIFAR-10

#### 4.2. Experimental Results on CIFAR-100

#### 4.3. Experimental Results on ImageNet

**Table 3.**Comparison with various compression methods for VGG-16, ResNet-18, and ResNet-50 on ImageNet dataset.

Model | Method | Compression Method | Top-5 Acc (%) | FLOPs Reduction |
---|---|---|---|---|

VGG-16 | AutoPruner [51] | Pruning | −1.49 | 3.79× |

Standard Tucker [10] | Tucker | −0.50 | 4.93× | |

HT-2 [52] | Tucker | −0.65 | 5.26× | |

Ours | CP | −0.38 | 5.26× | |

Resnet-18 | FPGM [53] | Pruning | −0.55 | 1.72× |

DSA [54] | Pruning | −0.73 | 1.72× | |

DACP [20] | Pruning | −1.48 | 1.89× | |

Standard Tucker [10] | Tucker | −1.55 | 2.25× | |

MUSCO [49] | Tucker | −0.30 | 2.42× | |

TRP [50] | Low-Rank Matrix | −2.34 | 2.60× | |

Ours | CP | −0.16 | 2.61× | |

Resnet-50 | TRP [50] | Low-Rank Matrix | −0.80 | 1.80× |

Standard Tucker [10] | Tucker | −1.75 | 2.04× | |

AKECP [55] | Pruning | −2.30 | 2.62× | |

HRANK [18] | Pruning | −1.86 | 2.64× | |

HT-2 [52] | Tucker | −0.71 | 2.85× | |

AutoPruner [51] | Pruning | −1.62 | 2.94× | |

Ours | CP | −0.51 | 2.97× |

#### 4.4. Discussion

#### 4.4.1. Sensitivity-Constrained Optimization

#### 4.4.2. Iterative Fine-Tuning Based on Sensitivity Order

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Convolution layer and its CP decomposition. Each transparent box corresponds to the three-way tensor $\mathcal{X}$, $\mathcal{Z}$, ${\mathcal{Z}}^{\prime}$, and $\mathcal{Y}$ in Equations (6)–(8), with frontal sides corresponding to spatial dimensions. Arrows represent linear mappings and illustrate how scalar values on the right are computed. Small boxes correspond to single elements of the target tensor. (

**a**) Original convolution layer. (

**b**) CP decomposition of the convolution layer. Yellow tube, blue box, and red tube correspond to 1 × 1, D × D, and 1 × 1 convolutions in (6), (7), and (8), respectively.

**Figure 2.**Overall workflow of the proposed iterative low-rank tensor decomposition based on the sensitivity of convolution layers.

**Figure 3.**Fine-tuning curves for ResNet-18 on ImageNet dataset after only CP decomposing convolutional layer 1 of block 4, with and without sensitivity-constrained optimization.

**Figure 4.**Performance evaluation of ResNet-18 on ImageNet after only CP decomposing convolutional layer 1 of block 4, with and without sensitivity-constrained optimization with various ranks.

**Table 1.**Comparison with various tensor decomposition methods for ResNet-20 and ResNet-32 on CIFAR-10 dataset.

Model | Method | Compression Method | Top-1 Acc (%) | Compression Ratio |
---|---|---|---|---|

Resnet-20 | Standard Tucker [10] | Tucker | −3.84 | 2.6× |

Standard Tensor Train [47] | Tensor Train | −4.55 | 5.4× | |

Standard Tensor Ring [48] | Tensor Ring | −3.75 | 5.4× | |

PSTR-S [46] | Tensor Ring | −0.45 | 2.5× | |

PSTR-M [46] | Tensor Ring | −2.75 | 6.8× | |

Ours | CP | −0.31 | 2.6× | |

Ours | CP | −1.82 | 6.8× | |

Resnet-32 | Standard Tucker [10] | Tucker | −4.79 | 5.1× |

Standard Tensor Train [47] | Tensor Train | −4.19 | 4.8× | |

Standard Tensor Ring [48] | Tensor Ring | −1.89 | 5.1× | |

PSTR-S [46] | Tensor Ring | −1.05 | 2.7× | |

PSTR-M [46] | Tensor Ring | −1.89 | 5.8× | |

Ours | CP | −0.45 | 2.8× | |

Ours | CP | −1.31 | 5.8× |

**Table 2.**Comparison with various tensor decomposition methods for ResNet-20 and ResNet-32 on CIFAR-100 dataset.

Model | Method | Compression Method | Top-1 Acc (%) | Compression Ratio |
---|---|---|---|---|

Resnet-20 | Standard Tucker [10] | Tucker | −7.87 | 2.5× |

Standard Tensor Train [47] | Tensor Train | −3.76 | 5.6× | |

Standard Tensor Ring [48] | Tensor Ring | −1.85 | 4.7× | |

PSTR-S [46] | Tensor Ring | +0.73 | 2.3× | |

PSTR-M [46] | Tensor Ring | −1.78 | 4.7× | |

Ours | CP | +1.05 | 2.6× | |

Ours | CP | −1.32 | 4.7× | |

Resnet-32 | Standard Tucker [10] | Tucker | −9.07 | 2.5× |

Standard Tensor Train [47] | Tensor Train | −5.20 | 4.6× | |

Standard Tensor Ring [48] | Tensor Ring | −1.40 | 4.8× | |

PSTR-S [46] | Tensor Ring | −0.05 | 2.4× | |

PSTR-M [46] | Tensor Ring | −1.33 | 5.2× | |

Ours | CP | −0.25 | 2.5× | |

Ours | CP | −0.81 | 5.2× |

Model | Method | Top-1 Acc (%) | Top-5 Acc (%) | FLOPs Reduction |
---|---|---|---|---|

Resnet-18 | CP-one-shot | −4.09 | −2.57 | 2.61× |

CP-random | −2.23 | −1.31 | 2.61× | |

Ours | −1.28 | −0.16 | 2.61× | |

Resnet-50 | CP-one-shot | −6.27 | −3.62 | 2.97× |

CP-random | −4.07 | −1.86 | 2.97× | |

Ours | −1.34 | −0.51 | 2.97× |

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**MDPI and ACS Style**

Yang, C.; Liu, H.
Stable Low-Rank CP Decomposition for Compression of Convolutional Neural Networks Based on Sensitivity. *Appl. Sci.* **2024**, *14*, 1491.
https://doi.org/10.3390/app14041491

**AMA Style**

Yang C, Liu H.
Stable Low-Rank CP Decomposition for Compression of Convolutional Neural Networks Based on Sensitivity. *Applied Sciences*. 2024; 14(4):1491.
https://doi.org/10.3390/app14041491

**Chicago/Turabian Style**

Yang, Chenbin, and Huiyi Liu.
2024. "Stable Low-Rank CP Decomposition for Compression of Convolutional Neural Networks Based on Sensitivity" *Applied Sciences* 14, no. 4: 1491.
https://doi.org/10.3390/app14041491