ExpFamilyPCA.jl

ExpFamilyPCA.jl is a Julia package for exponential family principal component analysis (EPCA), a versatile generalization of PCA designed to handle non-Gaussian data, enabling dimensionality reduction and data analysis across a wide variety of distributions (e.g., binary, count, and compositional data). It is designed for applications in machine learning (belief compression, text analysis), signal processing (denoising), and data science (sample debiasing, clustering, dimensionality reduction), but can be applied to other fields with diverse data types.

• Website: https://sisl.github.io/ExpFamilyPCA.jl/dev/
• Math: https://sisl.github.io/ExpFamilyPCA.jl/dev/math/intro/
• API Documentation: https://sisl.github.io/ExpFamilyPCA.jl/dev/api/

Features

• Implements exponential family PCA (EPCA)
• Supports multiple exponential family distributions
• Flexible constructors for custom distributions
• Fast symbolic differentiation and optimization
• Numerically stable scientific computation

Installation

To install the package, use the Julia package manager. In the Julia REPL, type:

using Pkg; Pkg.add("ExpFamilyPCA")

Supported Distributions

The following distributions are supported:

Distribution	Description
BernoulliEPCA	For binary data
BinomialEPCA	For count data with a fixed number of trials
ContinuousBernoulliEPCA	For probabilities between 0 and 1
GammaEPCA	For positive continuous data
GaussianEPCA	Standard PCA for real-valued data
NegativeBinomialEPCA	For over-dispersed count data
ParetoEPCA	For heavy-tailed distributions
PoissonEPCA	For count and discrete distribution data
WeibullEPCA	For life data and survival analysis

Quickstart

Each EPCA object supports the following methods:

• fit!: Trains the model and returns compressed training data.
• compress: Compresses new input data.
• decompress: Reconstructs original data from the compressed representation.

Example:

X = sample_from_poisson(n1, indim)
Y = sample_from_poisson(n2, indim)
epca = PoissonEPCA(indim, outdim)

X_compressed = fit!(epca, X)
Y_compressed = compress(epca, Y)
Y_reconstructed = decompress(epca, Y_compressed)

Custom Distributions

When working with custom distributions, certain specifications are often more convenient and computationally efficient than others. For example, inducing the gamma EPCA objective from the log-partition $G(\theta) = -\log(-\theta)$ and its derivative $g(\theta) = -1/\theta$ is much simpler than implementing the full the Itakura-Saito distance:

$$ D(P(\omega), \hat{P}(\omega)) =\frac{1}{2\pi} \int_{-\pi}^{\pi} \Bigg[ \frac{P(\omega)}{\hat{P}(\omega)} - \log \frac{P(\omega)}{\hat{P}{\omega}} - 1\Bigg] d\omega. $$

In ExpFamilyPCA.jl, we would write:

G(θ) = -log(-θ)
g(θ) = -1 / θ
gamma_epca = EPCA(indim, outdim, G, g, Val((:G, :g)); options = NegativeDomain())

A lengthier discussion of the EPCA constructors and math is provided in the documentation.

Contributing

Contributions are welcome! If you want to contribute, please fork the repository, create a new branch, and submit a pull request. Before contributing, please make sure to update tests as appropriate.