Similarity of Neural Network Representations Revisited (1905.00414v4)

Abstract: Recent work has sought to understand the behavior of neural networks by comparing representations between layers and between different trained models. We examine methods for comparing neural network representations based on canonical correlation analysis (CCA). We show that CCA belongs to a family of statistics for measuring multivariate similarity, but that neither CCA nor any other statistic that is invariant to invertible linear transformation can measure meaningful similarities between representations of higher dimension than the number of data points. We introduce a similarity index that measures the relationship between representational similarity matrices and does not suffer from this limitation. This similarity index is equivalent to centered kernel alignment (CKA) and is also closely connected to CCA. Unlike CCA, CKA can reliably identify correspondences between representations in networks trained from different initializations.

• The paper introduces centered kernel alignment (CKA) to address CCA's limitations, significantly enhancing neural network representation analysis.
• It validates CKA's performance through experiments, achieving nearly 99.3% accuracy in layer correspondence on the CIFAR-10 dataset.
• CKA's diagnostic capabilities support model debugging, transfer learning, and architectural optimizations across diverse neural network models.

Similarity of Neural Network Representations Revisited

The exploration of neural network representations has been pivotal in understanding the mechanisms driving deep learning models. In the paper "Similarity of Neural Network Representations Revisited" by Simon Kornblith, Mohammad Norouzi, Honglak Lee, and Geoffrey Hinton, the authors revisit methods for comparing neural network representations, primarily focusing on canonical correlation analysis (CCA). This work introduces the concept of centered kernel alignment (CKA) as a robust alternative, addressing inherent limitations in traditional CCA when applied to high-dimensional neural network activations.

Problem Statement and Background

The primary problem tackled in this paper is the assessment of similarities between neural network layers and across different models. Given matrices $X \in \mathbb{R}^{n \times p_1}$ and $Y \in \mathbb{R}^{n \times p_2}$ representing activations of $p_1$ and $p_2$ neurons for $n$ examples, the goal is to design a scalar similarity index $s(X, Y)$ that remains meaningful even as activation dimensions exceed the number of data points. Previous efforts using CCA and its variants often fall short in these scenarios due to their invariance to invertible linear transformations, which prove insufficient in practical neural network settings with high-dimensional data.

Centered Kernel Alignment (CKA)

CKA is introduced as a similarity index that measures the relationship between representational similarity matrices (RSMs). This measure is not only derivable from CCA but extends beyond it by maintaining efficacy in identifying correspondences across varied dimensional scales. The essence of CKA lies in its capability to grasp the shared subspace structure, weighted by eigenvalues representing the variance explained by the corresponding eigenvectors.

Key Contributions:

• Invariance Analysis: The authors argue that an effective similarity index should be invariant to orthogonal transformations and isotropic scaling but not to invertible linear transformations. This perspective shapes their critique of existing methods and motivates the development of CKA.
• CKA Formulation and Connection to SVCCA: The paper establishes the equivalence of CKA to centered kernel alignment and draws connections to linear regression and related methods. This equivalence underscores CKA's robustness in identifying layer correspondences even when models are trained with different initializations.
• Empirical Validation: Through extensive experiments on various neural network architectures, CKA is shown to consistently outperform other indexes such as CCA, SVCCA, and PWCCA, particularly in scenarios involving deep convolutional networks and Transformer models. The results indicate that CKA can reveal intricate relationships between layers that other methods fail to capture.

Numerical Results and Observations

The detailed experiments highlight several intriguing numerical results:

• Accuracy in Layer Correspondence Identification: CKA demonstrates almost perfect accuracy (99.3%) in identifying corresponding layers across models trained from different initializations on the CIFAR-10 dataset. This performance vastly surpasses other methods, with Linear Regression achieving only 45.4% and others such as CCA and SVCCA performing even worse.
• Efficacy in Different Network Depths and Widths: CKA effectively identifies redundant layers in overly deep networks, providing a diagnostic tool for detecting pathological behaviors. Additionally, it reveals that increasing the width of layers in neural networks leads to more similar representations across models, with the similarity of earlier layers saturating faster than later layers.
• Cross-Architecture Similarity: CKA adeptly captures representational similarities between layers of different architectures, such as Plain networks versus ResNets, something previously proposed methods found challenging.

Practical and Theoretical Implications

The implications of adopting CKA are significant for both practical applications and theoretical explorations:

• Cross-Model Transfer Learning: Understanding layer correspondences facilitated by CKA can enhance techniques for transfer learning, allowing for more efficient model fine-tuning and adaptation across different tasks.
• Model Debugging and Optimization: CKA's diagnostic capabilities can aid in identifying inefficiencies in model architectures, prompting architectural adjustments that can lead to better-performing models.
• Broader Validation: The robust validation framework proposed in this paper sets a new standard for evaluating similarity measurement methods, crucial for advancing research in neural network interpretability.

Future Directions

While CKA marks a significant advancement, several avenues remain open for exploration. Future work could involve:

• Kernel Design Improvements: Investigating alternative kernels beyond linear and RBF for CKA, potentially uncovering even more nuanced similarities between neural network layers.
• Scaling to Larger Datasets: Extending the applicability of CKA to massive datasets and more complex models, such as those used in large-scale LLMs or image recognition tasks.
• Integration with Unsupervised Learning: Combining CKA with unsupervised learning techniques to analyze and enhance unsupervised representations in neural networks.

Conclusion

The paper by Kornblith et al. significantly advances our understanding of neural network representation similarities by introducing CKA. Its robust framework and strong empirical performance offer a new lens through which to compare and diagnose neural network architectures, showcasing its capability to improve both model interpretability and performance.