Which Frequencies do CNNs Need? Emergent Bottleneck Structure in Feature Learning

2024-02-12

International Conference on Machine Learning

AI Summary

Convolutional neural networks (CNNs) empirically exhibit strong frequency biases and hierarchical processing, yet the structural principles underlying their representational efficiency remain poorly understood. Method: We introduce the convolutional bottleneck (CBN)—a naturally emergent architectural motif wherein early layers compress inputs into sparse frequency-channel representations, and later layers reconstruct outputs from this compressed representation. We define the CBN rank to quantify the number and types of critical frequencies preserved in the bottleneck, and integrate Fourier-domain analysis, parameter complexity theory, stability-driven derivations of activation/weight structures, and empirical validation. Contribution/Results: We prove that parameter norm scales linearly with network depth and CBN rank, tightly linking it to the regularity of the target function. Crucially, we provide the first optimization-efficiency–driven theoretical justification for downsampling: optimally parameter-efficient CNNs must possess a CBN structure. Our framework successfully decodes learned frequency preferences and functional mechanisms in multi-task CNNs, establishing a new paradigm for understanding CNN representation learning and designing efficient architectures.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Deep Neural Architectures and Foundation Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We describe the emergence of a Convolution Bottleneck (CBN) structure in CNNs, where the network uses its first few layers to transform the input representation into a representation that is supported only along a few frequencies and channels, before using the last few layers to map back to the outputs. We define the CBN rank, which describes the number and type of frequencies that are kept inside the bottleneck, and partially prove that the parameter norm required to represent a function $f$ scales as depth times the CBN rank $f$. We also show that the parameter norm depends at next order on the regularity of $f$. We show that any network with almost optimal parameter norm will exhibit a CBN structure in both the weights and - under the assumption that the network is stable under large learning rate - the activations, which motivates the common practice of down-sampling; and we verify that the CBN results still hold with down-sampling. Finally we use the CBN structure to interpret the functions learned by CNNs on a number of tasks.

Problem

• Explains emergence of Convolution Bottleneck structure in CNNs.
• Defines CBN rank to describe frequency retention in networks.
• Links parameter norm to function regularity and network depth.

Innovation

• Convolution Bottleneck (CBN) structure emergence
• CBN rank defines frequency and channel retention
• Parameter norm scales with depth and CBN rank