Convolutional Neural Networks are extremely efficient architectures in image and audio recognition tasks, thanks to their ability to exploit the local translational invariance of signal classes over their domain. In this paper we consider possible generalizations of CNNs to signals defined on more general domains without the action of a translation group. In particular, we propose two constructions, one based upon a hierarchical clustering of the domain, and another based on the spectrum of the graph Laplacian. We show through experiments that for low-dimensional graphs it is possible to learn convolutional layers with a number of parameters independent of the input size, resulting in efficient deep architectures.
核心贡献 · Key contributions
提出两种将 CNN 推广到图结构数据的方法：空间构建和谱构建。 Proposes two generalizations of CNNs to graph-structured data: spatial and spectral constructions.
表明空间构建通过层次聚类和局部感受野实现 O(n)参数。 Shows that spatial construction achieves O(n) parameters using hierarchical clustering and local receptive fields.
引入基于图拉普拉斯特征向量的谱构建，每个特征图实现 O(n)参数。 Introduces spectral construction using graph Laplacian eigenvectors, achieving O(n) parameters per feature map.
通过平滑谱乘子展示 O(1)参数构建，使参数数量与输入规模无关。 Demonstrates O(1) parameter construction via smooth spectral multipliers, enabling parameter count independent of input size.
在低维图数据集上验证两种构建，显示相比全连接网络效率提升。 Validates both constructions on low-dimensional graph datasets, showing improved efficiency over fully connected networks.
表明平滑谱约束产生空间局部化滤波器，可跨图位置泛化。 Shows that smooth spectral constraints yield spatially localized filters that generalize across graph locations.
局限 · Limitations
空间构建缺乏跨不同图位置的权重共享。 Spatial construction lacks weight sharing across different graph locations.
谱构建需要昂贵的特征向量及其转置乘法。 Spectral construction requires expensive multiplication by eigenvectors and their transpose.
高频特征向量可能含噪声；谱构建可能无法从中捕获有意义信息。 High-frequency eigenvectors may be noisy; spectral construction may not capture meaningful information from them.
空间构建中使用的朴素多尺度聚类不保证与图拉普拉斯的兼容性。 Naive multiscale clustering used in spatial construction does not guarantee compatibility with graph Laplacian.
实验限于低维图；未充分探索高维或大规模图的可扩展性。 Experiments limited to low-dimensional graphs; scalability to high-dimensional or large-scale graphs not fully explored.
论文章节 · Sections（共 18）
摘要Abstract
1 引言1 Introduction
1.1 贡献1.1 Contributions
2 空间构造2 Spatial Construction
2.1 通过 W 的局部性2.1 Locality via W𝑊W
2.2 图上的多分辨率分析2.2 Multiresolution Analysis on Graphs
2.3 深度局部连接网络2.3 Deep Locally Connected Networks
3 谱构造3 Spectral Construction
3.1 加权图上的调和分析3.1 Harmonic Analysis on Weighted Graphs
3.2 通过拉普拉斯谱扩展卷积3.2 Extending Convolutions via the Laplacian Spectrum
3.3 重新发现标准 CNN3.3 Rediscovering standard CNN’s
3.4 使用平滑谱乘子的 O(1)构造3.4 O(1)𝑂1O(1) construction with smooth spectral multipliers