Similarity of Neural Network Representations Revisited

By Simon Kornblith, Mohammad Norouzi, Honglak Lee, Geoffrey E. Hinton. 2019

2,348 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Representation & Self-Supervised Learning

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Representation & Self-Supervised Learning

Citations in the corpus

Unraveling Meta-Learning: Understanding Feature Representations for Few-Shot Tasks (2020, background): “For example, ImageNet pre-training of large ResNets has become an industry standard for transfer learning (Kornblith et al., 2019).”
Margin-Based Few-Shot Class-Incremental Learning with Class-Level Overfitting Mitigation (2022, methodology): “Finally, we compare the CKA similarity [21] for measuring the similarity between f simple and the f target .”
Transfer without Forgetting (2022, background): “1, left) significantly alters its parameters at all layers while tasks progress, resulting in a very low Centered Kernel Alignment [32] similarity score already at the first CL task.”
Probing Representation Forgetting in Supervised and Unsupervised Continual Learning (2022, background): “One signiﬁcant difference is that in [38], the authors use linear CKA (centred Kernel Alignment) [23] to measure the similarity between intermediate representations inﬂuenced by forgetting, while in our work we measure how much forgetting has occurred on the representations using LP.”
Representational Continuity for Unsupervised Continual Learning (2021, methodology): “We analyze the similarity between the representations learnt between (i) Two independent UCL models, (ii) Two independent SCL models (iii) SCL and UCL models using centered kernel alignment (CKA) (Kornblith et al., 2019) in Figure 3, which provides a score between 0 and 1 measuring the similarity be”
How Well Does Self-Supervised Pre-Training Perform with Streaming Data? (2021, methodology): “See Appendix C.3 for details of the CKA similarity.”
Linear Mode Connectivity in Multitask and Continual Learning (2020, methodology): “In particular we consider Euclidean distance and Central Kernel Alignment (CKA) (Kornblith et al., 2019).”
What is being transferred in transfer learning? (2020, background): “More speciﬁcally, we measure the (cid:96) 2 distance between 2 P-T s and 2 RI-T s, both per Interestingly, RI-T s are farther from each other compared to two P-T s,”
Anatomy of Catastrophic Forgetting: Hidden Representations and Task Semantics (2020, methodology): “Representational similarity techniques take in a pair of neural network (hidden) layer representations and output a similarity score between 0 and 1 depicting how similar the representations are [21, 22, 38].”
Rapid Learning or Feature Reuse? Towards Understanding the Effectiveness of MAML (2019, methodology): “Table 3 presents CCA and CKA (another representational similarity measure, from [15]) similarities for MAML and ANIL models.”
Emerging Cross-lingual Structure in Pretrained Language Models (2019, methodology): “Kornblith et al. (2019) argues that CCA fails to measure meaningful similarities between representations that have a higher dimen-sion than the number of data points and introduce the centered kernel alignment (CKA) to solve this problem.”
Compositional Few-Shot Class-Incremental Learning (2024, methodology): “Then, the similarity between X and Z y can be obtained as Recent studies have demonstrated that the linear CKA similarity exhibits reliability comparable to the kernel CKA similarity.”
Cross-Layer Distillation with Semantic Calibration (2020, methodology): “However, intermediate layers in teacher and student models tend to have different levels of abstraction [18], [20], [23].”
Similarity Analysis of Contextual Word Representation Models (2020, methodology): “A similar pattern was found in vision networks (Kornblith et al., 2019).”
Refiner: Refining Self-attention for Vision Transformers (2021, methodology): “The metric CKA is calculated with linear kernel function as proposed in [28].”
The Platonic Representation Hypothesis (2024, neutral): “Hence, the CKA score reflects the global similarities of the models.”
Do Self-Supervised and Supervised Methods Learn Similar Visual Representations? (2021, methodology): “[16], indicating that SL and SimCLR utilize the residual architecture in a similar way, with residual blocks decoupling from each other.”