# Neural Network based Explicit Mixture Models and Expectation-maximization based Learning

## TL;DR

This paper introduces two explicit neural network mixture models with analytical likelihood forms, developed using expectation-maximization algorithms, and demonstrates their effectiveness in sample generation and classification tasks.

## Contribution

The paper presents novel explicit mixture models based on flow neural networks and develops EM-based learning algorithms for these models, enhancing likelihood computation and sample generation.

## Key findings

- Models enable likelihood computation and efficient sampling.
- Flow-based neural networks used as generators.
- Models show improved sample quality and classification accuracy.

## Abstract

We propose two neural network based mixture models in this article. The proposed mixture models are explicit in nature. The explicit models have analytical forms with the advantages of computing likelihood and efficiency of generating samples. Computation of likelihood is an important aspect of our models. Expectation-maximization based algorithms are developed for learning parameters of the proposed models. We provide sufficient conditions to realize the expectation-maximization based learning. The main requirements are invertibility of neural networks that are used as generators and Jacobian computation of functional form of the neural networks. The requirements are practically realized using a flow-based neural network. In our first mixture model, we use multiple flow-based neural networks as generators. Naturally the model is complex. A single latent variable is used as the common input to all the neural networks. The second mixture model uses a single flow-based neural network as a generator to reduce complexity. The single generator has a latent variable input that follows a Gaussian mixture distribution. We demonstrate efficiency of proposed mixture models through extensive experiments for generating samples and maximum likelihood based classification.