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Matrix logarithm parametrizations for neural network covariance models.

Peter M. Williams1

  • 1School of Cognitive and Computing Sciences, University of Sussex, Falmer, Brighton, UK

Neural Networks : the Official Journal of the International Neural Network Society
|March 29, 2003
PubMed
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This study addresses challenges in neural network modeling of probability distributions. It explores matrix-logarithm parametrization for covariance matrices, ensuring valid parameters and investigating prior invariance for predictive distributions.

Area of Science:

  • Machine Learning
  • Statistics
  • Computational Neuroscience

Background:

  • Neural networks commonly model conditional probability distributions by linking network outputs to distributional parameters.
  • A key challenge is ensuring network outputs satisfy constraints on distributional parameters, such as positive definiteness for covariance matrices.

Purpose of the Study:

  • To explore the matrix-logarithm parametrization for covariance matrices in multivariate normal distributions.
  • To investigate the relationship between parameter choices, prior distributions, and predictive distributions in Bayesian inference.

Main Methods:

  • Utilizing neural networks to represent distributional parameters as functions of conditioning events.
  • Applying matrix-logarithm transformations to ensure covariance matrices are positive definite.

Related Experiment Videos

  • Analyzing the invariance of predictive distributions under different priors for the chosen parametrization.

    • Main Results:

    • The matrix-logarithm parametrization provides a valid method for incorporating covariance matrix constraints within neural networks.
    • The study demonstrates how this parametrization impacts the choice of priors and the resulting predictive distributions.
    • Invariance properties of predictive distributions were investigated with respect to specific classes of priors.

    Conclusions:

    • The matrix-logarithm approach offers a robust solution for modeling constrained covariance matrices in neural networks.
    • Understanding the interplay between parametrization and priors is crucial for effective Bayesian modeling with neural networks.
    • This work contributes to more principled and flexible neural network-based probabilistic modeling.