Insights on representational similarity in neural networks with canonical correlation (1806.05759v3)

Abstract: Comparing different neural network representations and determining how representations evolve over time remain challenging open questions in our understanding of the function of neural networks. Comparing representations in neural networks is fundamentally difficult as the structure of representations varies greatly, even across groups of networks trained on identical tasks, and over the course of training. Here, we develop projection weighted CCA (Canonical Correlation Analysis) as a tool for understanding neural networks, building off of SVCCA, a recently proposed method (Raghu et al., 2017). We first improve the core method, showing how to differentiate between signal and noise, and then apply this technique to compare across a group of CNNs, demonstrating that networks which generalize converge to more similar representations than networks which memorize, that wider networks converge to more similar solutions than narrow networks, and that trained networks with identical topology but different learning rates converge to distinct clusters with diverse representations. We also investigate the representational dynamics of RNNs, across both training and sequential timesteps, finding that RNNs converge in a bottom-up pattern over the course of training and that the hidden state is highly variable over the course of a sequence, even when accounting for linear transforms. Together, these results provide new insights into the function of CNNs and RNNs, and demonstrate the utility of using CCA to understand representations.

• The paper introduces Projection Weighted CCA to effectively differentiate between signal and noise in neural network representations.
• It demonstrates that networks generalizing from true labels converge to more similar internal representations than those that memorize, with wider networks exhibiting greater similarity.
• The study reveals that RNNs follow a bottom-up convergence pattern with significant temporal variation in hidden states during training.

Overview of Canonical Correlation Analysis in Neural Networks

The paper "Insights on representational similarity in neural networks with canonical correlation" by Morcos et al. explores the application of Canonical Correlation Analysis (CCA) as a tool to compare and understand neural network representations. As neural networks operate with increasingly complex structures, understanding how internal representations evolve and compare across different networks is crucial but remains challenging. This paper builds on previous work with SVCCA and introduces Projection Weighted CCA to address the differentiation between signal and noise in neural network representations.

Key Contributions

1. Projection Weighted CCA: The authors identify a key limitation in the existing methodologies, namely the inability to effectively distinguish between signal and noise. Projection Weighted CCA improves upon SVCCA by considering the importance of CCA vectors through projection weighting. This method more accurately assesses the representational similarity by accounting for signal contributions over mere noise, as demonstrated in various experimental setups including synthetic data comparisons.
2. Comparative Analysis Across CNNs and RNNs: Using the refined CCA approach, the paper reveals that networks which generalize (i.e., trained on true labels) tend to converge to more similar internal representations than networks that memorize data. The analysis extends to demonstrate that wider networks exhibit a greater convergence to similar solutions compared to narrower counterparts. Among CNNs with the same topology but varying learning rates, distinct clusters with diverse representations emerge, showcasing the parameter sensitivity in training dynamics.
3. Insights on RNN Training Dynamics: The research also explores representational dynamics within RNNs, showing that RNNs tend to converge in a bottom-up pattern during training. Notably, the hidden state varies significantly over time, suggesting high sensitivity to sequence history and input variability. This conclusion points toward distinctly nonlinear representational shifts in RNNs across sequence timesteps, raising further inquiries into the nuanced learning processes in sequential models.

Numerical Results and Implications

The results are quantitatively significant; for instance, the projection weighting robustly handles noise sensitivity across varying signal-to-noise ratios in synthetic tests. By utilizing the improved methodology, the researchers provide concrete examples where similar architectural choices and parameter changes impact representational similarity in surprising ways.

From a practical perspective, these insights underscore the importance of nuanced architectural choices and hyperparameter settings in designing neural networks, particularly for tasks requiring robust generalizations from training data. The projection-weighted approach can be seen as a foundational tool for further studies aiming to elucidate complex learning dynamics in deep networks.

Future Directions

This work opens multiple avenues for future research. There is an interest in deeply understanding the properties of neuron directions that preserve across differing network initializations, as well as leveraging learned similarities as potential regularizers for improving model generalization. Additionally, exploring dynamic training strategies akin to freeze training for RNNs could yield advanced techniques in handling complex sequence tasks. Such explorations could significantly refine the theoretical understanding of deep neural networks and their training dynamics, pushing the boundaries of artificial intelligence further.

In conclusion, the paper makes a substantial contribution to the computational understanding of neural networks' representational capabilities, providing a strong methodological foundation for examining and improving learning models across variable conditions and settings.