Deep Skew-t Mixture Models
Abstract
High-dimensional clustering is challenging when component distributions are both heavy-tailed and directionally asymmetric. We propose a deep skew-t mixture model (DStMM), a hierarchical factor-analytic mixture based on the generalised-hyperbolic skew-t normal mean--variance representation. A shared inverse-gamma mixing variable is propagated along each complete latent pathway, allowing heavy tails and directional asymmetry to be modelled jointly while preserving conditional Gaussianity. Each complete pathway therefore admits an exact GHST marginal representation. We formalise the reductions to symmetric deep t, Gaussian deep-mixture, and single-layer GHST factor-analytic models, discuss local non-identifiability and the implementation-level parameter-counting convention, and derive the conditional generalised inverse Gaussian law used for estimation. Estimation is carried out by a stochastic/Monte Carlo EM algorithm, with an explicit implementation-based parameter count for BIC architecture comparison. Simulation studies show that DStMM performs similarly to the symmetric robust model when skewness is absent but provides increasing gains as directional asymmetry becomes stronger, particularly under heavier tails; the same qualitative behaviour persists under smaller samples and unequal mixture proportions. Two real-data applications provide complementary evidence. On the UCI handwritten-digit benchmark, DStMM gives the strongest clustering performance under a common deep architecture, while on the Gas Sensor Array Drift data, DStMM improves on both deep Gaussian and deep t alternatives and, under the implemented BIC criterion, selects a non-trivial second mixture layer. Together, these results support the value of propagating skewness and heavy-tail variation through a deep latent mixture while retaining an exact pathway-level likelihood.
Related
- Sandwich Variance Estimation in Scale Mixture of Skew-Normal Linear Mixed Models.
- Classification and enumeration of skew morphisms of skew-type four on cyclic $2$-groups
- Classification of skew left braces with additive group isomorphic to the infinite dihedral group
- Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits
- The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring
BibTeX
@misc{wu2026deep,
title = {{Deep Skew-t Mixture Models}},
author = {Jinran Wu and You-Gan Wang and Geoffrey J. McLachlan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00773},
eprint = {2609.00773},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00773},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Deep Skew-t Mixture Models
AU - Jinran Wu
AU - You-Gan Wang
AU - Geoffrey J. McLachlan
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00773
UR - https://arxiv.org/abs/2609.00773
AB - High-dimensional clustering is challenging when component distributions are both heavy-tailed and directionally asymmetric. We propose a deep skew-t mixture model (DStMM), a hierarchical factor-analytic mixture based on the generalised-hyperbolic skew-t normal mean--variance representation. A shared inverse-gamma mixing variable is propagated along each complete latent pathway, allowing heavy tails and directional asymmetry to be modelled jointly while preserving conditional Gaussianity. Each complete pathway therefore admits an exact GHST marginal representation. We formalise the reductions to symmetric deep t, Gaussian deep-mixture, and single-layer GHST factor-analytic models, discuss local non-identifiability and the implementation-level parameter-counting convention, and derive the conditional generalised inverse Gaussian law used for estimation. Estimation is carried out by a stochastic/Monte Carlo EM algorithm, with an explicit implementation-based parameter count for BIC architecture comparison. Simulation studies show that DStMM performs similarly to the symmetric robust model when skewness is absent but provides increasing gains as directional asymmetry becomes stronger, particularly under heavier tails; the same qualitative behaviour persists under smaller samples and unequal mixture proportions. Two real-data applications provide complementary evidence. On the UCI handwritten-digit benchmark, DStMM gives the strongest clustering performance under a common deep architecture, while on the Gas Sensor Array Drift data, DStMM improves on both deep Gaussian and deep t alternatives and, under the implemented BIC criterion, selects a non-trivial second mixture layer. Together, these results support the value of propagating skewness and heavy-tail variation through a deep latent mixture while retaining an exact pathway-level likelihood.
ER -