Learning to Compress: Local Rank and Information Compression in Deep Neural Networks (2410.07687v1)

Abstract: Deep neural networks tend to exhibit a bias toward low-rank solutions during training, implicitly learning low-dimensional feature representations. This paper investigates how deep multilayer perceptrons (MLPs) encode these feature manifolds and connects this behavior to the Information Bottleneck (IB) theory. We introduce the concept of local rank as a measure of feature manifold dimensionality and demonstrate, both theoretically and empirically, that this rank decreases during the final phase of training. We argue that networks that reduce the rank of their learned representations also compress mutual information between inputs and intermediate layers. This work bridges the gap between feature manifold rank and information compression, offering new insights into the interplay between information bottlenecks and representation learning.

• The paper introduces local rank as a novel metric to quantify feature manifold dimensionality in neural networks.
• The paper demonstrates that local rank reduces during training, evidencing inherent information compression in learned representations.
• The paper connects local rank reduction with Information Bottleneck theory, suggesting efficient removal of redundant information.

Learning to Compress: Local Rank and Information Compression in Deep Neural Networks

This paper advances the understanding of representation learning in deep neural networks by examining the low-rank bias these models display during training, connecting it with Information Bottleneck (IB) theory. The authors introduce the novel concept of "local rank," quantifying the feature manifold's dimensionality and providing significant insights into the interplay between rank and mutual information compression.

Key Contributions

1. Definition and Analysis of Local Rank: Local rank is introduced as a metric for feature manifold dimensionality within neural networks. The paper presents theoretical findings on local rank behavior during training, linking it to inherent regularization effects that promote low-rank solutions.
2. Empirical Evidence of Rank Reduction: The authors demonstrate through experiments on synthetic and real-world datasets that local rank reduces in the final training stages. This rank reduction indicates that neural networks inherently compress their learned representations' dimensionality.
3. Connection to Information Bottleneck Theory: The paper explores the correlation between local rank reduction and mutual information compression, suggesting that a decrease in local rank aligns with the principles of the Information Bottleneck, where redundant information is minimized while retaining relevance to the output.

Theoretical Insights

Leveraging the Data Manifold Hypothesis, the authors propose that datasets often inhabit manifolds of much lower dimensionality than the ambient input space. Deep neural networks seem to implicitly capture these low-dimensional manifolds through their training process, guided by gradient descent toward low-rank weight solutions.

Propositions in this paper provide formalizations of conditions under which neural networks reduce the local rank, aligning with implicit regularization theories. Specifically, the implicit regularization minimizes weight matrix norms, leading to low-rank solutions across intermediate layers, effectively serving as bottlenecks.

Empirical Validation

Empirical studies employed both synthetic Gaussian datasets and the MNIST digit dataset to validate the theoretical results. The experimental outcomes consistently show that local rank diminishes during the final phase of training across various layers. Such observations affirm the theoretical assertions of the network's capacity to compress feature manifolds as training converges.

Information Theoretic Implications

In aligning local rank with the Information Bottleneck theory, the paper posits that the reduction in rank equates to an efficient compression of information. This is crucial for understanding how deep networks balance compression against prediction accuracy. For Gaussian variables, they demonstrate that varying the IB trade-off parameter results in discernible changes in the dimension of the learned representations, supporting theoretical expectations.

Implications and Future Directions

This exploration into local rank and information compression in neural networks offers significant implications for both theoretical modeling and practical application. By understanding the conditions under which networks compress information, new strategies for model optimization, such as targeted regularization and layer design, may emerge.

Future research could extend these findings to non-Gaussian settings and various network architectures beyond MLPs. Additionally, practical applications in efficient model deployment, such as compressing redundant parameters for faster inference and reduced computational overhead, present valuable avenues for exploration.

This paper provides a robust framework to further investigate and model the dynamics of deep learning, enhancing our understanding of how neural networks effectively encode and compress information within their architectures.