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Adaptive Graph Auto-Encoder for General Data Clustering
Xuelong Li, Hongyuan Zhang, Rui Zhang
TL;DR
General clustering often lacks a graph, limiting the direct use of graph convolution. AdaGAE builds and adaptively updates a weighted graph from learned embeddings, and experiments report significant performance and support the adaptive update.
Problem
General clustering data lack a provided graph, although graph-based methods capture manifold information useful for non-Euclidean data.
Source: §1 · p. 1Method
Results
Experiments report significant AdaGAE performance and ablations verify the effectiveness of adaptive graph updates.
Source: §5 · p. 7, §4.3 · p. 7Takeaways & Limitations
AdaGAE is reported stable across datasets, while adaptive updates and the weighted-graph architecture improve performance in the reported experiments.
Source: §4.3 · p. 7The approach assumes sparse weighted adjacency contains sufficient information for same-cluster samples to become more cohesive in latent space.
Source: §3.3 · p. 5
Abstract
from arXiv · showhide
Graph-based clustering plays an important role in the clustering area. Recent studies about graph convolution neural networks have achieved impressive success on graph type data. However, in general clustering tasks, the graph structure of data does not exist such that the strategy to construct a graph is crucial for performance. Therefore, how to extend graph convolution networks into general clustering tasks is an attractive problem. In this paper, we propose a graph auto-encoder for general data clustering, which constructs the graph adaptively according to the generative perspective of graphs. The adaptive process is designed to induce the model to exploit the high-level information behind data and utilize the non-Euclidean structure sufficiently. We further design a novel mechanism with rigorous analysis to avoid the collapse caused by the adaptive construction. Via combining the generative model for network embedding and graph-based clustering, a graph auto-encoder with a novel decoder is developed such that it performs well in weighted graph used scenarios. Extensive experiments prove the superiority of our model.
1 INTRODUCTION
Graph-based clustering captures manifold information, but general clustering often lacks a supplied graph, and CNN operations may not suit many data types. AdaGAE extends graph auto-encoding to general data with an adaptively updated graph and a strategy to avoid collapse.
1 INTRODUCTION
- Graph-based clustering captures manifold information for non-Euclidean data, whereas CNN-based clustering may not apply to text, social-network, signal, and data-mining datasets.The paper frames graph construction as important because general clustering data do not come with a graph.Source: §1 · p. 1
- Network embedding maps graph nodes to latent features for downstream node classification, clustering, and link prediction, with generative methods modeling connectivity distributions.The introduction situates graph auto-encoders among methods that extend neural networks to irregular graph data.Source: §1 · p. 1
- AdaGAE extends graph auto-encoding to general data by constructing a weighted graph generatively and updating it from learned embeddings.Its contributions include a weighted-graph decoder and a strategy to avoid collapse during graph updates.Source: §1 · p. 1
- AdaGAE alternates weighted-graph GAE training with graph updates from the learned embedding, increasing sparsity until convergence.The framework begins at sparsity k0 and updates with a larger k.Source: Figure 1 · p. 2, §1 · p. 2
2 PRELIMINARY AND RELATED WORK
The preliminaries define graph and data notation, review deep-clustering approaches, and describe graph convolution and graph auto-encoders, including their use of adjacency information.
2.1 Notations
- The paper denotes a graph as G = (V, E, W), data points as rows of X, and the number of points and clusters as n and c.Vectors and matrices use lowercase and uppercase letters, respectively.Source: §2.1 · p. 2
2.2 Deep Clustering
- Deep-clustering work includes auto-encoder-based methods and SpectralNet, while CNN-based clustering models are limited to image datasets and are not treated as main competitors.SpectralNet simulates spectral decomposition but does not investigate constructing a high-quality graph from deep representations.Source: §2.2 · p. 2
2.3 Graph Auto-Encoder
- A GCN layer applies normalized adjacency to node features, equivalent to weighted means over each node and its first-order neighbors; GAE learns embeddings and reconstructs input structure or content.Prior general-data work cited here manually constructs a graph and keeps it fixed during training.Source: §2.3 · pp. 2–3
3 PROPOSED MODEL
AdaGAE constructs a weighted graph from learned connectivity distributions and trains a graph auto-encoder whose graph is updated during clustering. Its adaptive sparsity schedule is designed to prevent graph degeneration and collapse while incorporating higher-level information.
3.1 Probabilistic Perspective of Weighted Graphs
- Conditional connectivity probabilities define weighted, generally directed links; a sparse distribution is estimated by balancing feature-space distance with regularization.The k-sparse construction is motivated by keeping links across clusters near zero and has k as its stated hyperparameter.Source: §3.1 · p. 3
3.2 Graph Auto-Encoder for Weighted Graph
- The graph auto-encoder uses the estimated connectivity distribution as its reconstruction target, with adaptive self-loops and a decoder that applies SoftMax to negative latent-feature distances.The loss also includes a local consistency penalty balanced against cross entropy by λ.Source: §3.2 · pp. 3–4
3.3 Adaptive Graph Auto-Encoder
- Updating the graph is intended to exploit higher-level information and correct links based on low-level relationships, but a simple embedding-based update can collapse.With fixed sparsity, the graph can degenerate into an unweighted graph; the proposed response is to increase k during updates to reconnect sub-clusters before they scatter.Source: §3.3 · pp. 4–5
3.3.1 Another Explanation of Degeneration
- With fixed k, better reconstruction can produce a worse graph update, so perfect reconstruction may yield poor performance.Theorem 2 interprets the decoder’s SoftMax output from negative distances as an optimization involving expected distance and KL divergence.Source: §3.3.1 · p. 5
3.4 Spectral Analysis
- Adaptive self-loops reduce the spectrum of the normalized Laplacian, with the largest nonzero eigenvalue increasing from λ_n to λ′_n.The passage states 0 = λ_1 = λ′_1 < λ_n < λ′_n.Source: §3.4 · p. 5
3.5 Computational Complexity
- Graph construction takes O(n^2) time, while clustering assignments take O(n^2c) with spectral clustering or O(ndc) with k-means.The passage also gives O(|E|d_i) per GCN update iteration.Source: §3.5 · p. 5
4 EXPERIMENTS
AdaGAE is evaluated across text, UCI, and image datasets against classical, deep-clustering, and graph-auto-encoder baselines. The experiments report stable performance across datasets and examine adaptive updates, sparsity, weighted graphs, and the effect of λ.
4.1 Datasets and Compared Methods
- The evaluation covers 10 text, UCI, and image datasets and compares AdaGAE with 11 classical, deep-clustering, and GAE-based methods.USPS is a subset of 1,854 samples, and features are rescaled to [0, 1].Source: §4.1 · pp. 5–6
4.2 Experimental Setup
- ACC and NMI are reported as means over 10 runs; the setup searches λ from 10^-3 to 10^3 and uses 10 graph-update steps.The encoder has two GCN layers, and the maximum GAE optimization iterations vary from 150 to 200.Source: §4.2 · p. 6
4.3 Experimental Results
- The reported ACC and NMI comparisons appear in Tables 1 and 3, respectively.The supplied passages do not provide the tables' cell values.Source: §4.3 · p. 6, Table 1 · p. 6, Table 3 · p. 7
- AdaGAE is stable across all datasets, while classical clustering performs poorly at scale and existing GAE-based models fail on general datasets.The authors suggest the latter may reflect algorithmically constructed graphs containing only low-level information.Source: §4.3 · p. 7
- On most datasets, fixed sparsity causes AdaGAE to collapse; on COIL20, ACC shrinks about 50% and NMI shrinks about 40%.The authors explain that adaptive increases in sparsity connect groups before degeneration, making within-cluster embeddings more cohesive.Source: §4.3 · p. 7
- The adaptive graph update plays a positive role, and the weighted-graph architecture improves performance on most datasets.The USPS visualization shows increasingly cohesive embeddings across 10 graph-update epochs.Source: §4.3 · pp. 6–7
- The second loss term is especially important on UMIST, while a λ that is too large can produce a trivial embedding.AdaGAE performs well when λ is not too large.Source: Figure 4 · p. 6, §4.3 · p. 7
5 CONCLUSION
The conclusion presents AdaGAE as a general-data clustering model that learns a weighted graph and updates it adaptively. It links dynamic sparsity to avoiding collapse and reports experimental and visualization support.
5 CONCLUSION
- AdaGAE combines a generative graph representation model for weighted graphs with a graph auto-encoder for general-data clustering.The learned embedding helps adaptively update the artificially constructed graph to exploit high-level information.Source: §5 · p. 7
- The authors' theoretical analyses explain why fixed sparsity collapses during graph updates and argue that dynamic sparsity is essential.Source: §5 · p. 7
- Experiments and ablations report significant performance and support the effectiveness of adaptive updates, while visualizations support the theoretical analysis.Source: §5 · p. 7
6 PROOFS
The proofs address three theorems through connectivity-distribution reasoning, a constrained optimization derivation, and an argument modeled on prior work.
6.1 Proof of Theorem 1
- Theorem 1's proof considers a node's connectivity distribution and the effect of updating it according to distances.It states that the proof extends to the other nodes.Source: §6.1 · pp. 7–8
6.2 Proof of Theorem 2
- Theorem 2's proof rewrites the problem as an individual subproblem, then uses a Lagrangian and KKT conditions.Source: §6.2 · p. 8
6.3 Proof of Theorem 3
- Theorem 3's proof follows the corresponding argument in and invokes a lemma analogous to Lemma 3 there.Source: §6.3 · p. 8