SpectralNet: Spectral Clustering using Deep Neural Networks (1801.01587v6)

Abstract: Spectral clustering is a leading and popular technique in unsupervised data analysis. Two of its major limitations are scalability and generalization of the spectral embedding (i.e., out-of-sample-extension). In this paper we introduce a deep learning approach to spectral clustering that overcomes the above shortcomings. Our network, which we call SpectralNet, learns a map that embeds input data points into the eigenspace of their associated graph Laplacian matrix and subsequently clusters them. We train SpectralNet using a procedure that involves constrained stochastic optimization. Stochastic optimization allows it to scale to large datasets, while the constraints, which are implemented using a special-purpose output layer, allow us to keep the network output orthogonal. Moreover, the map learned by SpectralNet naturally generalizes the spectral embedding to unseen data points. To further improve the quality of the clustering, we replace the standard pairwise Gaussian affinities with affinities leaned from unlabeled data using a Siamese network. Additional improvement can be achieved by applying the network to code representations produced, e.g., by standard autoencoders. Our end-to-end learning procedure is fully unsupervised. In addition, we apply VC dimension theory to derive a lower bound on the size of SpectralNet. State-of-the-art clustering results are reported on the Reuters dataset. Our implementation is publicly available at https://github.com/kstant0725/SpectralNet .

• The paper presents a deep learning approach that learns a parameterized spectral embedding to overcome traditional clustering limitations.
• It integrates a Siamese network to learn complex affinities, enhancing clustering accuracy beyond standard Gaussian methods.
• Empirical results on benchmarks like MNIST and Reuters demonstrate state-of-the-art performance and efficient generalization to new data.

An Overview of SpectralNet: Spectral Clustering using Deep Neural Networks

The paper introduces SpectralNet, an innovative approach to spectral clustering that leverages deep neural networks to tackle scalability and out-of-sample extension issues commonly faced by traditional spectral clustering methods. Spectral clustering is renowned for its ability to handle non-convex clusters and optimize pairwise distances; however, it struggles with large datasets due to the computational intensity of eigen-decomposition and lacks a direct way to generalize spectral embeddings to unseen data points.

Key Contributions

SpectralNet merges deep learning with spectral clustering by learning a parameterized map from input data to its spectral embedding in the eigenspace of the graph Laplacian matrix. The map is trained using a constrained stochastic optimization procedure, which facilitates scalability—a critical improvement over standard spectral clustering methods that are limited by the cost of directly computing eigenvectors for large datasets. Additionally, the learned map enables out-of-sample-extension (OOSE) by allowing the trained network to compute spectral embeddings for new data points efficiently.

Methodology

The architecture of SpectralNet comprises two main components:

1. Spectral Mapping using a Neural Network: The network learns to map data points to their spectral embedding coordinates. This transformation is achieved through a series of feed-forward layers and a specialized output layer enforcing orthogonality, critical for maintaining the geometric properties required for effective spectral clustering.
2. Affinity Learning through Siamese Networks: To enhance clustering fidelity, the paper proposes using a Siamese network to learn affinities from unlabeled data instead of relying solely on Gaussian affinities computed from Euclidean distances. This affinity learning step captures complex relationships between data points and improves clustering outcomes.

The paper emphasizes that the stochastic training of SpectralNet is foundational for scaling the approach to larger datasets. It aligns with current trends in deep learning, where stochastic approximation is frequently used to handle voluminous and high-dimensional data.

Empirical Results and Theoretical Insights

Experiments conducted on benchmark datasets, including MNIST and Reuters, demonstrate that SpectralNet achieves clustering accuracy comparable to or exceeding state-of-the-art algorithms, such as DEC, DCN, and VaDE. Notably, on the Reuters dataset, SpectralNet achieves state-of-the-art results, underscoring its robustness and applicability to different data modalities.

The authors also investigate the theoretical underpinnings of SpectralNet through VC dimension analysis, which provides a bound on the size of neural networks required to represent the spectral clustering function class. This analysis highlights that the expressiveness of spectral clustering captured by SpectralNet is non-trivial and requires networks with a number of parameters growing linearly with the dataset size, emphasizing the complexity and capacity needed to approximate Laplacian eigenvectors.

Implications and Future Directions

The combination of spectral methods and neural networks in SpectralNet exemplifies a broader trend in machine learning: integrating classical algorithms with deep learning to overcome their traditional limitations. Practically, SpectralNet's ability to generalize spectral clustering to unseen data points without recomputing the full affinity matrix extends its usability to dynamic datasets often encountered in real-world applications, such as streaming data environments and large-scale data analytics.

Future work could explore extending SpectralNet to handle multi-view or heterogeneous data sources, leveraging transfer learning techniques to further power its predictive capability across related datasets. The framework opens the door to incorporating other graph-based learning problems into a scalable neural network-driven framework, potentially broadening its reach within unsupervised learning and beyond.