Gaussian Mixture Models in Pytorch

Implements gaussian mixture models in pytorch. Loss is computed with respect to mean negative log likelihood and optimized via gradient descent.

Usage

Run demo

usage: demo.py [-h] [--samples SAMPLES] [--components COMPONENTS] [--dims DIMS]
               [--iterations ITERATIONS]
               [--family {full,diagonal,isotropic,shared_isotropic,constant}] [--log_freq LOG_FREQ]
               [--radius RADIUS] [--mixture_lr MIXTURE_LR] [--component_lr COMPONENT_LR]
               [--visualize VISUALIZE] [--seed SEED]

Fit a gaussian mixture model to generated mock data

options:
  -h, --help            show this help message and exit
  --samples SAMPLES     The number of total samples in dataset
  --components COMPONENTS
                        The number of gaussian components in mixture model
  --dims DIMS           The number of data dimensions
  --iterations ITERATIONS
                        The number optimization steps
  --family {full,diagonal,isotropic,shared_isotropic,constant}
                        Model family, see `Mixture Types`
  --log_freq LOG_FREQ   Steps per log event
  --radius RADIUS       L1 bound of data samples
  --mixture_lr MIXTURE_LR
                        Learning rate of mixture parameter (pi)
  --component_lr COMPONENT_LR
                        Learning rate of component parameters (mus, sigmas)
  --visualize VISUALIZE
                        True for visualization at each log event and end
  --seed SEED           seed for numpy and torch

Usage

data = load_data(...)

model = GmmFull(num_components=3, num_dims=2)

loss = fit_model(
    model,
    data,
    num_iterations=10_000,
    mixture_lr=1e-5
    component_lr=1e-2
)

# visualize
print(f"Final Loss: {loss:.2f}")
plot_data_and_model(data, model)

Run tests

python3 -m pytest tests

Derivation

We start with the probability density function of a multivariate gaussian parameterized by mean $\mu \in \mathbb{R}^{d}$ and the covariance matrix $\Sigma \in \mathrm{S}_+^d$. The PDF describes the likelihood of sampling a point $x\in\mathbb{R}^{d}$ from the distribution.

$$\mathcal{N}(\mathbf{x}) = \frac{1}{(2\pi)^{k/2}|\Sigma|^{1/2}} \exp\left(-\frac{1}{2} (\mathbf{x} - \boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x} - \boldsymbol{\mu})\right)$$

In order to describe a mixture of gaussians, we add an additional parameter $\pi_k \in \Delta^{k-1}$ which assigns the probability that a sample comes from any of the $K$ gaussian components.

$$p(\mathbf{x}) = \sum_{k=1}^{K} \pi_k \mathcal{N}(\mathbf{x} | \boldsymbol{\mu}_k, \boldsymbol{\Sigma}_k)$$

Given elements of a dataset $x \in X^{(D \times N)}$, we want our model to fit to the data. This means maximizing the likelihood that the elements could have been sampled from the mixture PDF $p(\mathbf{x})$. Applying the function $-log(p(\mathbf{x}))$ for each element $\mathbf{x}$ has the effect of lowerbounding the best possible probability (1) while leaving the cost of an unlikely point (~0) unbounded. Given a dataset which contains few outliers and sufficently many components to cover the dataset, these properties make the negative log likelihood a suitable choice for our objective function.

$$f(\mathbf{x}) = - \frac{1}{N} \sum_{i=1}^{N} \ln{ \sum_{k=1}^{K} \pi_k \mathcal{N}(\mathbf{x}_i | \boldsymbol{\mu}_k, \boldsymbol{\Sigma}_k) }$$

For a from-scratch implementation of negative log likelihood backpropogation, see GMMScratch

Gaussian Model Types

Type	Description
Full	Fully expressive eigenvalues. Data can be skewed in any direction
Diagonal	Eigenvalues align with data axes. Dimensional variance is independent
Isotropic	Equal variance in all directions. Spherical distributions
Shared	Equal variance in all directions for all components
Constant	Variance is not learned and is equal across all dimensions and components

While more expressive varieties are able to better fit to real-world data, they require learning more parameters and are often less stable during training. As of now only Full, Diagonal, and Isotropic mixture types have been implemented.

Comparison to Expectation Maximization (EM) Algorithm

For more information, see On Convergence Properties of the EM Algorithm for Gaussian Mixtures.

Singularity Mitigation

From Pattern Recognition and Machine Learning by Christopher M. Bishop, pg. 433:

Suppose that one of the components of the mixture model, let us say the jth component, has its mean μ_j exactly equal to one of the data points so that μ_j = x_n for some value of n. If we consider the limit σ_j → 0, then we see that this term goes to infinity and so the log likelihood function will also go to infinity. Thus the maximization of the log likelihood function is not a well posed problem because such singularities will always be present and will occur whenever one of the Gaussian components ‘collapses’ onto a specific data point.

A common solution to this problem is to reset the mean of the offending component whenever a singularity appears. In practice, singularities can be mitigated by clamping the minimum value of elements on the covariance diagonal. In a stochastic environment, a large enough clamp value will allow the model to recover after a few iterations.