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Deep Unsupervised Clustering with Gaussian Mixture Variational Autoencoders
Nat Dilokthanakul, Pedro A. M. Mediano, Marta Garnelo, Matthew C. H. Lee, Hugh Salimbeni, Kai Arulkumaran, Murray Shanahan
TL;DR
The paper addresses unsupervised clustering with deep generative models while confronting over-regularisation, which can cause cluster degeneracy in Gaussian-mixture VAEs. It introduces a VAE with a Gaussian-mixture prior, applies a minimum information constraint, and reports competitive clustering with distinct, interpretable clusters.
Problem
Unsupervised clustering with deep generative models requires a model that represents latent structure while avoiding the over-regularisation effects known in VAEs.
Method
The paper uses a Gaussian-mixture prior in a VAE, constructs an inference model for direct reparameterization-based optimisation, and applies a minimum information constraint.
Results
The model achieves competitive unsupervised-clustering results, with learned clusters that are distinct, interpretable, and associated with meaningful visible-data features.
Takeaways & Limitations
Sampling shows that images generated from the same latent cluster share relevant high-level features while training remains entirely unsupervised.
Takeaways & Limitations
Future deep GMVAEs for hierarchical clustering must still address enduring optimisation challenges associated with VAEs.
Abstract
from arXiv · showhide
We study a variant of the variational autoencoder model (VAE) with a Gaussian mixture as a prior distribution, with the goal of performing unsupervised clustering through deep generative models. We observe that the known problem of over-regularisation that has been shown to arise in regular VAEs also manifests itself in our model and leads to cluster degeneracy. We show that a heuristic called minimum information constraint that has been shown to mitigate this effect in VAEs can also be applied to improve unsupervised clustering performance with our model. Furthermore we analyse the effect of this heuristic and provide an intuition of the various processes with the help of visualizations. Finally, we demonstrate the performance of our model on synthetic data, MNIST and SVHN, showing that the obtained clusters are distinct, interpretable and result in achieving competitive performance on unsupervised clustering to the state-of-the-art results.
1 INTRODUCTION
Unsupervised clustering remains difficult because classical methods capture local data-space relations, while separate dimensionality reduction and clustering impose independent assumptions. The paper proposes clustering within the VAE framework using a multimodal prior and relates this approach to latent disentanglement and related deep generative models.
- Classical k-means and Gaussian mixture models are limited to local data-space relations and cannot capture hidden hierarchical dependencies in latent spaces.
- Applying classical clustering after deep-generative dimensionality reduction is unsatisfactory because the two techniques generally use independent assumptions.
- The paper proposes unsupervised clustering within the VAE framework by assuming observed data arise from a multimodal prior distribution.
- The proposed inference model is directly optimisable using the reparameterization trick, while over-regularisation can severely affect clustering performance.
- The work relates unsupervised clustering to disentangling latent variables and differs from M1+M2 by targeting unsupervised clustering rather than semi-supervised classification.
- The paper compares its results with deep embedded clustering, adversarial autoencoders, and categorial GANs.
2 VARIATIONAL AUTOENCODERS
Variational autoencoders combine variational Bayesian inference with neural-network flexibility and scalability. They make otherwise intractable inference tractable through neural-network posteriors and the reparametrization trick, enabling stochastic-gradient optimisation and backpropagation.
- VAEs combine variational Bayesian methods with neural networks to provide flexible and scalable inference models.
- Neural networks output the conditional posterior, allowing the variational-inference objective to be optimised with stochastic gradient descent and standard backpropagation.
- The reparametrisation trick enables backpropagation through continuous stochastic variables without directly relying on Monte Carlo methods.
3 GAUSSIAN MIXTURE VARIATIONAL AUTOENCODERS
The GMVAE replaces the usual unimodal Gaussian prior with a neural-network-parameterized Gaussian mixture, enabling latent-space clustering while retaining tractable reparameterized inference. Its ELBO combines reconstruction and prior-related terms, but the discrete-variable regularizer can merge clusters through over-regularization.
- A Gaussian mixture prior extends the regular VAE’s unimodal latent representation to support distinct latent-space classes.Inferring a data point’s mixture component corresponds to inferring its latent class.
- The generative model represents x|w as a K-component Gaussian mixture whose means and variances are produced by a neural network.A uniformly distributed one-hot z selects the mixture component, while w conditions the component parameters.
- The observation model generates y from continuous latent x using a neural network, and the variational family factorizes into Gaussian recognition models for x and w plus the derived z-posterior.The z-posterior is computed from x and w rather than introduced as an independently sampled variational factor.
- The ELBO contains reconstruction, conditional-prior, w-prior, and z-prior terms, with Monte Carlo estimation and standard reparameterization used where needed.The model construction avoids sampling discrete z when estimating the conditional-prior contribution.
- The z-prior regularizer can reduce divergence by moving encoded points and mixture means together, encouraging cluster overlap and degenerate assignments.This regularization is expected to become overpowering as the amount of training data increases.
4 EXPERIMENTS
Experiments show that GMVAE can model non-Gaussian synthetic data and learn meaningful image clusters, but over-regularisation can collapse latent clusters; the minimum information constraint mitigates this failure. On MNIST and SVHN, the learned representations are differentiated, interpretable, and competitive for unsupervised clustering.
- The experiments examine optimisation dynamics on synthetic data, MNIST clustering, and generated images conditioned on latent variables.
- Synthetic data: Over-regularisation merges latent clusters, making every data point equally likely to belong to any cluster and erasing class information.The z-prior can dominate early training, driving the model toward a poor local optimum.
- Synthetic data: The minimum information constraint suppresses the z-prior initially, allowing clusters to spread before later merging only sufficiently overlapping clusters.The modified objective holds the z-prior cost at a threshold until its cost becomes high enough.
- Synthetic data: GMVAE represents non-Gaussian synthetic densities more richly and accurately than a regular GMM with restrictive Gaussian assumptions.
- Unsupervised image clustering: On MNIST, cluster degeneracy does not arise, and GMVAE achieves classification scores competitive with state-of-the-art techniques except adversarial autoencoders.
- Unsupervised image clustering: Increasing the numbers of clusters and Monte Carlo samples makes GMVAE more robust and stable, whereas smaller settings can lead to poor local minima.
- Image generation: MNIST Gaussian components correspond to digit classes while w controls digit style, and SVHN components group visually similar images.
5 CONCLUSION
The paper introduces GMVAEs with a Gaussian-mixture latent level, addresses cluster degeneracy from VAE over-regularisation using standard heuristics, and evaluates the approach on clustering and generated images.
- GMVAEs place one latent encoding level in the form of a Gaussian mixture model and provide a variational Bayes optimisation objective.
- Over-regularisation manifests as cluster degeneracy in GMVAEs, but standard heuristics can solve this specific problem.
- The model achieves competitive results against the current state of the art on unsupervised clustering tasks using popular datasets.
- Images generated from the same latent cluster share relevant high-level visible features, including the same MNIST digit, despite unsupervised training.
- Stacking GMVAEs could support hierarchical clustering, but enduring VAE optimisation challenges remain important for this future direction.
A NETWORK PARAMETERS
The experiments use Adam with specified optimisation settings and neural architectures for inference, Gaussian-mixture latent conditionals, and dataset-specific observation models.
- Optimisation uses Adam with learning rate 10^-4, β1 = 0.9, β2 = 0.999, and ϵ = 10^-8.
- The qφ(x,w) inference network shares hidden layers before splitting into four output streams for x and w means and variances.
- The MNIST and SVHN experiments use convolutional architectures with dataset-specific input sizes and listed channel configurations.
- The pβ(x|w,z) architecture splits its output into 2K streams, with K streams for means and K for cluster variances.
- The pθ(y|x) observation model outputs Gaussian parameters for synthetic data and Bernoulli parameters for MNIST and SVHN.