Dimensionality Reduction as Probabilistic Inference

TL;DR

This paper introduces ProbDR, a unified probabilistic framework that interprets various classical dimensionality reduction algorithms as inference methods, enabling better uncertainty reasoning and model composition.

Contribution

The paper presents ProbDR, a novel variational framework that unifies many DR algorithms as probabilistic inference, facilitating reasoning about uncertainties and extensions.

Findings

ProbDR encompasses PCA, t-SNE, UMAP, and others as inference algorithms.

The framework enables reasoning about unseen data and uncertainties.

ProbDR allows integration with probabilistic programming languages.

Abstract

Dimensionality reduction (DR) algorithms compress high-dimensional data into a lower dimensional representation while preserving important features of the data. DR is a critical step in many analysis pipelines as it enables visualisation, noise reduction and efficient downstream processing of the data. In this work, we introduce the ProbDR variational framework, which interprets a wide range of classical DR algorithms as probabilistic inference algorithms in this framework. ProbDR encompasses PCA, CMDS, LLE, LE, MVU, diffusion maps, kPCA, Isomap, (t-)SNE, and UMAP. In our framework, a low-dimensional latent variable is used to construct a covariance, precision, or a graph Laplacian matrix, which can be used as part of a generative model for the data. Inference is done by optimizing an evidence lower bound. We demonstrate the internal consistency of our framework and show that it enables the use of probabilistic programming languages (PPLs) for DR. Additionally, we illustrate that the framework facilitates reasoning about unseen data and argue that our generative models approximate Gaussian processes (GPs) on manifolds. By providing a unified view of DR, our framework facilitates communication, reasoning about uncertainties, model composition, and extensions, particularly when domain knowledge is present.