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DeepCSHAP: Utilizing Shapley Values to Explain Deep Complex-Valued Neural Networks

Florian Eilers, Xiaoyi Jiang

TL;DR

The paper addresses the lack of explainability tools for complex-valued neural networks (CVNNs) by introducing a complex-domain SHAP framework, DeepC SHAP, and adapting gradient-based explanations to the complex setting via Wirtinger calculus. It develops a complex chain rule and a max-pooling SHAP variant, along with an open-source PyTorch library and evaluations on MNIST and PolSAR that demonstrate superior explanation quality for DeepC SHAP compared with gradient baselines. The work validates key SHAP properties such as local accuracy and missingness in CVNN explanations and provides theoretical proofs and runtime analyses. By enabling principled interpretation of CVNNs, the approach enhances transparency and reliability for applications involving real and complex data, including MRI and PolSAR imagery.

Abstract

Deep Neural Networks are widely used in academy as well as corporate and public applications, including safety critical applications such as health care and autonomous driving. The ability to explain their output is critical for safety reasons as well as acceptance among applicants. A multitude of methods have been proposed to explain real-valued neural networks. Recently, complex-valued neural networks have emerged as a new class of neural networks dealing with complex-valued input data without the necessity of projecting them onto $\mathbb{R}^2$. This brings up the need to develop explanation algorithms for this kind of neural networks. In this paper we provide these developments. While we focus on adapting the widely used DeepSHAP algorithm to the complex domain, we also present versions of four gradient based explanation methods suitable for use in complex-valued neural networks. We evaluate the explanation quality of all presented algorithms and provide all of them as an open source library adaptable to most recent complex-valued neural network architectures.

DeepCSHAP: Utilizing Shapley Values to Explain Deep Complex-Valued Neural Networks

TL;DR

The paper addresses the lack of explainability tools for complex-valued neural networks (CVNNs) by introducing a complex-domain SHAP framework, DeepC SHAP, and adapting gradient-based explanations to the complex setting via Wirtinger calculus. It develops a complex chain rule and a max-pooling SHAP variant, along with an open-source PyTorch library and evaluations on MNIST and PolSAR that demonstrate superior explanation quality for DeepC SHAP compared with gradient baselines. The work validates key SHAP properties such as local accuracy and missingness in CVNN explanations and provides theoretical proofs and runtime analyses. By enabling principled interpretation of CVNNs, the approach enhances transparency and reliability for applications involving real and complex data, including MRI and PolSAR imagery.

Abstract

Deep Neural Networks are widely used in academy as well as corporate and public applications, including safety critical applications such as health care and autonomous driving. The ability to explain their output is critical for safety reasons as well as acceptance among applicants. A multitude of methods have been proposed to explain real-valued neural networks. Recently, complex-valued neural networks have emerged as a new class of neural networks dealing with complex-valued input data without the necessity of projecting them onto . This brings up the need to develop explanation algorithms for this kind of neural networks. In this paper we provide these developments. While we focus on adapting the widely used DeepSHAP algorithm to the complex domain, we also present versions of four gradient based explanation methods suitable for use in complex-valued neural networks. We evaluate the explanation quality of all presented algorithms and provide all of them as an open source library adaptable to most recent complex-valued neural network architectures.
Paper Structure (22 sections, 25 equations, 6 figures, 3 tables)

This paper contains 22 sections, 25 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Explanation methods
  4. Complex-valued neural networks
  5. Real-valued explaination methods
  6. Gradient based methods
  7. DeepSHAP
  8. Complex-valued Gradient based methods
  9. Deep$\mathbb{C}$SHAP
  10. $\mathbb{C}$SHAP values
  11. Chain rule for Deep$\mathbb{C}$SHAP
  12. Max$\mathbb{C}$SHAP
  13. Evaluation
  14. MNIST experiment
  15. PolSAR experiment
  16. ...and 7 more sections

Figures (6)

  • Figure 1: Two different interpretations of ReLU and their corresponding complex-valued adaptions. $\mathbb{C}_n$ denotes the $n$-th quadrant.
  • Figure 2: Qualitative results for MNIST experiment. Explanation scores for output 8 and 3 are shown and the image after masking 20% of the most important pixels for both predictions. All images are normalized in itself, normalization over all methods is not suitable due to different orders of magnitude.
  • Figure 3: Quantitative results on MNIST experiment. Change in log-odds after masking 20% of the most relevant features for classes 8 and 3 according to different explanation algorithms. The median is shown in red.
  • Figure 4: Example image from the PolSAR S1SLC$\_$CVDL dataset. Magnitude and Phase information of the complex-valued two channel (HH and HV) images are shown.
  • Figure 5: Quantitative results for the PolSAR experiment. Fraction of calculated contributions from the correct channels versus all channels are visualized for all explanation methods introduced in this paper. For gradient based methods $|\cdot|$ denotes the absolute value of the respecting method, R+I denotes adding up real and imaginary part.
  • ...and 1 more figures