Source-linked AI summary
Deep Gaussian Mixture Models
Cinzia Viroli, Geoffrey J. McLachlan
TL;DR
Deep clustering has received less attention than deep supervised learning, motivating a multilayer Gaussian-mixture framework. The paper defines DGMMs, addresses identifiability and parameter estimation, and reports strong clustering performance in simulated and real-data experiments. It also identifies limitations from non-identifiability and from uncertainty propagation in layerwise estimation.
Problem
Deep methods for clustering have received limited research compared with deep methods for supervised tasks.
Method
The paper defines deep Gaussian mixtures as multilayer latent-variable models with Gaussian mixtures at each layer, and develops identifiability constraints and a unified stochastic estimation algorithm.
Results
DGMM achieves the best classification performance among the compared methods on the Smiley data and gives the best clustering method on the Ecoli data.
Takeaways & Limitations
DGMM generalizes several classical clustering models and is particularly suitable for complex data with large sample size because of its greater flexibility.
Takeaways & Limitations
Layerwise estimation can propagate sampling uncertainty and bias through subsequent layers, with the problem increasing with model depth, component count, and unbalanced paths.
Abstract
from arXiv · showhide
Deep learning is a hierarchical inference method formed by subsequent multiple layers of learning able to more efficiently describe complex relationships. In this work, Deep Gaussian Mixture Models are introduced and discussed. A Deep Gaussian Mixture model (DGMM) is a network of multiple layers of latent variables, where, at each layer, the variables follow a mixture of Gaussian distributions. Thus, the deep mixture model consists of a set of nested mixtures of linear models, which globally provide a nonlinear model able to describe the data in a very flexible way. In order to avoid overparameterized solutions, dimension reduction by factor models can be applied at each layer of the architecture thus resulting in deep mixtures of factor analysers.
1 Introduction
Deep learning has succeeded in supervised classification, but deep methods for clustering remain comparatively underexplored. This paper presents deep Gaussian mixtures for clustering and discusses their identifiability, estimation, and empirical performance.
- Deep learning uses multilayer architectures that learn many parameters through successive nonlinear transformations.
- Deep neural networks have achieved strong results in supervised classification, including face identification at 97.35% accuracy.
- Deep learning architectures are used for speech recognition, object detection, pattern recognition, image processing, and other supervised classification tasks.
- Research on deep methods for clustering has been limited despite the success of deep models for supervised tasks.
- The paper introduces deep Gaussian mixtures for clustering, discusses identifiability, proposes a stochastic estimation algorithm, and evaluates the models on simulated and real data.
2 Gaussian Mixture Models
Finite Gaussian mixture models are established tools for model-based clustering across many application areas. Their component distributions can also be represented through latent factor models with structured covariance decompositions.
- Finite mixture models are widely used for model-based clustering in pattern recognition, data mining, image analysis, and machine learning.
- A Gaussian mixture model represents a p-dimensional quantitative vector as a mixture with k Gaussian components.
- The Gaussian component parameters include mixing probabilities, means, and covariance matrices.
- A Gaussian mixture has a factor-analytic representation using latent standard-Gaussian variables and independent Gaussian errors with diagonal covariance matrices.
3 Deep Mixture Models
A DGMM is a multilayer network of latent variables whose layerwise Gaussian mixtures form nested linear models and a globally flexible nonlinear model. The architecture supports shared paths and dimension reduction, but requires constraints because component configurations and latent parameters can be non-identifiable.
- 3 Deep Mixture Models: A DGMM is a network of latent-variable layers in which each layer follows a Gaussian mixture, producing nested mixtures of linear models.Together, the layers provide a globally nonlinear model for flexible data description.
- 3.1 Definition: The model uses first-order Markov conditional structure: each layer depends conditionally on the next layer rather than higher layers.Formally, f(z(l)|z(l+1), ..., z(h); Θ) = f(z(l)|z(l+1); Θ).
- 3.1 Definition: Parameter tying shares sub-component parameters and reduces the computational cost relative to directly learning a model with k = 18 components.
- 3.2 Model-based clustering and identifiability: Without further restrictions, DGMMs have identifiability problems caused by layer component counts and equivalent paths, including models with the same total number of paths.For h = 2, configurations (k1, k2) = (2, 3) and (3, 2) both yield k = 6 paths and may be indistinguishable.
- 3.2 Model-based clustering and identifiability: A second identifiability issue concerns latent-factor parameter estimation, which is addressed through dimensional and diagonality constraints.The paper imposes a diagonality constraint on Λ⊤Ψ^-1Λ and analogous constraints at each DGMM layer.
- 3.2 Model-based clustering and identifiability: DGMMs generalize classical mixtures, mixtures of mixtures, and mixtures of factor analyzers, while deep mixtures of factor analyzers use progressively reduced latent dimensions.The constraint p > r1 > r2 > ... > rh ≥ 1 also helps avoid over-parameterized models, especially when p is high.
4 Deep dimensionally reduced Gaussian mixture models
Dimensionally reduced DGMMs use latent variables with progressively decreasing dimensions to control model complexity while encompassing several existing mixture-based clustering models.
- Dimension reduction: Dimension reduction uses sequential latent layers with dimensions p > r1 > r2 > . . . > rh ≥ 1.The parameter dimensions change accordingly across layers.
- Dimension reduction: A two-layer dimensionally reduced DGMM has latent variables of dimensions r1 and r2, with p > r1 > r2 ≥ 1.
- Model relationships: The framework generalizes Gaussian mixtures and mixtures of factor analyzers by varying the number of layers and applying dimension reduction.Gaussian mixtures arise without layers or dimension reduction, while mixtures of factor analyzers correspond to a one-layer model.
- Model relationships: Special parameter settings recover mixtures with common factor loadings, heteroscedastic factor mixture analysis, and mixtures of factor mixture analyzers.The latter is described as a two-layer deep mixture with k1 > 1.
- Estimation motivation: A unified estimation algorithm is proposed because greedy layer-wise learning propagates latent-sampling bias through subsequent layers, worsening with depth, component count, and unbalanced paths.The cited prior approach learns one layer at a time and then uses posterior samples as data for the next layer.
5 Fitting Deep Gaussian Mixture Models
DGMM fitting uses EM-style latent-variable estimation, but the hierarchical posterior calculations are costly; a stochastic EM strategy samples latent variables layer by layer to make the updates tractable.
- EM formulation: EM is natural for DGMM estimation because the model is hierarchical, alternating conditional-expectation and parameter-maximization steps.The E- and M-steps operate on the complete-data log-likelihood evaluated at provisional parameters.
- Computational challenge: Unlike classical GMMs, DGMMs require multivariate latent-variable posteriors, making direct estimation very slow and unsuitable for large datasets.Classical GMM posterior computation involves only the allocation variable for each component.
- Stochastic estimation: SEM or MCEM addresses the computational difficulty by drawing latent-variable observations or samples from their conditional density during the E-step.These stochastic variants simplify the expectation calculation by replacing direct posterior integration with sampling.
- Stochastic estimation: The algorithm samples pseudorandom latent observations sequentially at each layer from f(z^(l)|z^(l−1), s; Θ′), starting with z^(0) = y.Variables at the upper model level are fixed for the current parameter fit.
- Layer-wise updates: The hierarchical generation structure allows the stochastic E- and M-steps to be computed separately for each layer.For layers l = 1, . . . , h, the algorithm maximizes the corresponding objective using the sampled latent variables.
- Posterior calculations: Posterior allocation probabilities f(s_l|y_i) are computed using Bayes’ formula within the estimation procedure.
6 Simulated and Real Application
The paper evaluates DGMM clustering on simulated Smiley data and several real datasets, comparing it with classical, mixture-based, and factor-analyzer methods. DGMM achieves the best or preferred clustering results across several tasks, including the Smiley, Ecoli, Vehicle, and Satellite analyses.
- Smiley Data: The Smiley simulation generated 1,000 observations from four classes in three dimensions, with two clustering-relevant variables and one Gaussian noise variable.The relevant variables formed two Gaussian eyes, a triangular nose, and a parabolic mouth.
- Smiley Data: DGMM achieved the best classification performance on Smiley data across 100 replicates, evaluated using Adjusted Rand Index and misclassification rate.The comparison fixed the number of groups at the true value and reported average indicators with standard errors.
- Real Data: The real-data evaluation compared DGMM with Gaussian, skewed, k-means, medoid, hierarchical, factor-mixture, and mixture-of-factor-analyzers methods.The number of groups was assumed known to compare clustering performance across methods.
- Real Data: On Ecoli data, DGMM was the best clustering method despite eight highly unbalanced classes, while SNmm and STmm failed to converge.The selected DGMM used r1 = 2, r2 = 1 and k1 = 8, k2 = 1.
- Real Data: DGMM performed better on the difficult Vehicle task, using progressive dimension reduction with r1 = 7, r2 = 1 and components k1 = 4, k2 = 3.Vehicle classes were especially difficult to distinguish among the cars.
- Real Data: For Satellite data, a three-layer DGMM was preferred by BIC, although MFA had slightly higher ARI and fewer corrected classified units overall.The DGMM used dimensions r1 = 13, r2 = 2, r1 = 1 and components k1 = 6, k2 = 2, k1 = 1 as reported.
7 Final remarks
The paper presents DGMM as a broad framework for unsupervised classification that generalizes several mixture models. Experiments on simulated and real data report efficient clustering with two or three layers, with model choice guided by information criteria.
- DGMM provides a general framework for unsupervised classification encompassing classical mixtures, mixtures of mixtures, and mixtures of factor analyzers.
- Its greater flexibility comes with higher complexity, making DGMM particularly suitable for data with large sample size.
- Experiments on simulated and real data found efficient clustering performance with h = 2 and h = 3 layers.
- Information criteria can be used to choose the model configuration.